So the response needs to be on several different levels. First of all, you want to know about truth vs. the Bible. We Christians care deeply about both, and see them as partially intersecting circles. The Bible is true, but there is more to truth than is in the Bible (science, for instance). We also understand that truth is more a person than a thing. Jesus IS truth (God is truth), and truth as a philosophical/scientific construct is a reflection of the nature of God and the foundation of existence than it is a separate entity. Since the Bible is the revelation of God and his contract with humanity, it is true, and belief in the Bible is not at odds with caring about truth.

Secondly, I'm quite disappointed that you couldn't find the information you were after with a brief Google search. Again we find out the inadequacy of knowledge by Google alone. Exodus 34 is not meant to be a rewriting of the original laws on fresh tablets. God commands Moses to chisel *two* stone tablets. Chapter 34 is a renewal of the covenant not meant to be the Ten Commandments. After renewing and reiterating the covenant (v. 27), which Moses was to write down, in v. 28 Moses was to write on the tablets the Ten Commandments. There is no necessary assumption that the covenant expressed in vv. 10-26 was the same "ten words" that Moses was to write on the tablets. In Ex. 20, after the 10 commandments are expressed (vv. 2-17), other commands are given that are not part of the 10 (vv. 23-26). Then in chapters 21-23 (known as the book of the covenant), more laws are given that are not part of the 10. And, interestingly, there are many parts of chapter 23 that correspond to Ex. 34.10-26:

1. Let the fields lie unplowed during each 7th year (Ex. 23.10-11; 34.21)
2. Celebrate the Feast of Unleavened Bread (Ex. 23.15; 34.18)
3. "No one is to appear before me empty-handed (Ex. 23.15; 34. 20)
4. Celebrate the Feast of Harvest (Ex. 23.16; 34.22)
5. Celebrate the Feast of Ingathering (Ex. 23.16; 34.22)
6. Don't offer blood and yeast (Ex. 23.18; 34.25)
7. Bring the first-fruits of the soil (Ex. 23.19; 34.26)
8. I will drive out the people groups (Ex. 23.23; 34.11)
9. Don't bow down and worship their gods (Ex. 23.24; 34.14)
10. Break down their altars (23.24; 34.13)
11. I will enlarge your territory (23.31: 34.24)

What makes the most sense is that the 10 Commandments written in chapter 34 (though they are not spelled out in 34) are the same as the ones in Ex. 20, and that the words of the covenant Moses is writing in 34.10-26 (though not as fully spelled out) are the same as the words of the covenant Moses is writing in Ex. 21-23. Both are 40-day writing periods, and there is enough similarity between the situations and chapters 20-23 and chapter 34 to understanding that this is what is going on.

As far as "what is in the ark of the covenant," Exodus 25.16 says that Moses was to put in it "the tablets of the covenant law." It was a common practice in the ancient Near East to house law codes in specially constructed containers to represent their presentation before deity. What is placed in the ark, then, is most likely the written record of the covenantal agreements YHWH has made with his people. It may be the Ten Commandments, but it may be more than that. The terminology doesn't specify. It may be the "tablets of stone" of Ex. 24.12, but there is no certainty what all is written on those tablets. The tradition is that it is the Ten Commandments. The original set was destroyed (Ex. 32.19), then replaced by God (Ex. 34.1), along with other covenantal directives, and something of it was housed in the ark (Dt. 10.5). We don't know exactly what, but that's not being intellectually dishonest or twisting the truth.

In other words, nothing is being shoved under the rug, and we do care about truth as much as we honor belief in the Bible. We aren't guilty of intellectual dishonesty, we aren't ignoring anything, and we haven't twisted anything.

Exodus 34:27-28 demonstrates that Christianity cares more about belief in the Bible than about truth. Introduction

I know this post has become a really long wall of text, but please hear me out. There is a really big inconsistency in the Bible in Exodus 34:27-28. This doesn't have to be a problem -- there are plenty of other contradictions that are quickly explained away—but this one is so big that addressing it would do more harm than good for someone's faith. As a result, it is pretty much impossible to find an adequate explanation of this, which I think demonstrates that Bible scholars care more about having people believe in Christianity than having people know the truth about the Bible.

The inconsitency

So, let's first get some context. Let's start with Moses going onto the mountain Sinai. This happens in Exodus 19 and right away in Exodus 20, God gives Moses the 10 commandments we all know and love. This is also the passage you will be linked to if you google the 10 commandments. After this, you get a really long boring part where God instructs all sorts of laws—how to treat slaves, the measurements of the ark, all that kind of jazz. Where it gets interesting again is at Exodus 32 where we get the story of the golden calf. You'll probably be familiar with this story, but I'll summarise: Moses is on mount Sinai for 40 days and 40 nights. In the meantime, the Israelites build a calf of gold, which they start worshipping. When Moses comes down and sees them worshipping the calf, he gets really angry and smashes the tablets with the 10 commandments in his anger. So, now there is a problem: the tablets that were supposed to go into the ark of the covenant are broken, resulting in Moses having to go back up mount Sinai again and asking God for the 10 commandments again. "Fine," says God, in Exodus 34, "here are the 10 commandments again:"

"Do not worship any other god, for the Lord, whose name is Jealous, is a jealous God. Be careful not to make a treaty with those who live in the land; for when they prostitute themselves to their gods and sacrifice to them, they will invite you and you will eat their sacrifices. And when you choose some of their daughters as wives for your sons and those daughters prostitute themselves to their gods, they will lead your sons to do the same. Do not make any idols. Celebrate the Festival of Unleavened Bread. For seven days eat bread made without yeast, as I commanded you. Do this at the appointed time in the month of Aviv, for in that month you came out of Egypt. The first offspring of every womb belongs to me, including all the firstborn males of your livestock, whether from herd or flock. Redeem the firstborn donkey with a lamb, but if you do not redeem it, break its neck. Redeem all your firstborn sons. No one is to appear before me empty-handed. Six days you shall labor, but on the seventh day you shall rest; even during the plowing season and harvest you must rest. Celebrate the Festival of Weeks with the firstfruits of the wheat harvest, and the Festival of Ingathering at the turn of the year. Three times a year all your men are to appear before the Sovereign Lord, the God of Israel. I will drive out nations before you and enlarge your territory, and no one will covet your land when you go up three times each year to appear before the Lord your God. Do not offer the blood of a sacrifice to me along with anything containing yeast, and do not let any of the sacrifice from the Passover Festival remain until morning. Bring the best of the firstfruits of your soil to the house of the Lord your God. Do not cook a young goat in its mother’s milk." (Exodus 34:14-26)

Now, I'm pretty sure that these are not the same 10 commandments as the first time around—I think I would have remembered something as hilarious as not being allowed to cook a young goat in it's mother's milk being one of the laws we base our society on ;). So, you might ask yourself, are these really the 10 commandments? Does the text really say that? Thankfully, the Bible is pretty clear on that in the following two verses:

Then the Lord said to Moses, “Write down these words, for in accordance with these words I have made a covenant with you and with Israel. Moses was there with the Lord forty days and forty nights without eating bread or drinking water. And he wrote on the tablets the words of the covenant—the Ten Commandments." (Exodus 34:27-28)

Later on, in Exodus 40, Moses puts these two tablets in the ark and that's pretty much where the story ends.

Christian explanations

Naturally, I found this a pretty interesting thing I didn't know about the Bible when I was a Christian, so I figured I might as well do a little research what Christians had to say about this. Turns out that's pretty much nothing.

I first simply googled "what's in the ark of the covenant?". This should give some answers on what commandments were on the tablets, but it didn't. Of the top 6 results I got that mentioned the 10 commandments, 3 of them only said the phrase "10 commandments" with no further explanation which commandments (1, 2, 3), and 3 did include a reference or link, but referenced to Exodus 20 instead of Exodus 34 (1, 2, 3), implying that either they put the smashed tablets into the ark or that the 10 commandments on the second set of tablets were identical to the first one, which, as you have just read for yourself, they obviously aren't.

Now, I think this is pretty dishonest, but maybe I'm judging too quickly. After all, these sources are intended to give quick information about the ark, not a whole bible study. So, I figured I might as well try to find a more in depth commentary on the passage, that perhaps might settle these questions. After googling "Exodus 34 explained", that optimism quickly faded. Of my top 5 hits, 2 don't even mention the fact that these laws are referred to as the 10 commandments (1, 2), 2 do mention the 10 commandments, but fail to address the fact that the commandments given are not even close to the initial 10 commandments (1, 2), and only one (1) actually addresses it (kudos to Jamieson, Fausset and Brown). The way it is addressed isn't really that convincing—they say that Moses wrote these commandments and God wrote the 10 commandments onto the same slabs, which obviously isn't what the text says—but I do applaud the effort. However, the trend on how scholars deal with the subject remains to ignore the issue, sadly enough. Even all the commentaries on BibleHub about this specific verse don't mention the fact that these 10 commandments are not even close to the original commandments.

Conclusion

By now, I think it has become painfully obvious that Christians would rather not address this issue. This cannot be because it's not an important issue. You would expect that the fact that they might have been putting up the wrong set of 10 commandments in courthouses and schools and refer to as the basis for our morality and western society, is be somewhat of a problem. So, the only other reason I can think of for this issue not being addressed is intellectual dishonesty. I think that the people writing Bible studies are well aware of the problem, but even more aware of the fact that this problem doesn't have good answers. As a result, their best option, if their ultimate goal is having people believe the Bible to be the inspired word of God, even if that means twisting the truth, is to simply ignore the issue and move on. If they really did care about truth, they would either describe the issue and say that they don't know how to address it or give it a try to address it, even if that means some people will stop believing in the Bible.

Ultimately, Christianity doesn't care about truth. It only cares about belief in the Bible.