I'm no expert, but my understanding has always been that God does interact with people, and in a sense does in fact affect history.
There's evidence of this in the miracles in people's daily lives or the prophets themselves. An example off the top of my head is when The Prophet PBUH was alone with Abu Bakr in the desert when they had to hide in a cave, and God covered the entrance with a spider's web and a bird's nest to give the impression that no one could have gone in. I mention this miracle/mo'geza as an example because that was God interacting/intervening with The Prophet beyond the main mo'geza of The Qur'an.

Actually, Allah does interact with people in one way or another. The story of Moses is also in our Qur'an, and we all know how Allah helped Moses in it. Also, supplicating to our lord in Islam is pretty much asking Allah to enter in our lives and either give us what we ask for or save us from terrible punishments, even if it's not done in a "divine" manner.

> "Am I right in thinking that Allah does not personally enter into human history and act as a historical agent, that He always deals with the world through his word, prophets, and angels?"

Firstly, Allah, Exalted is He, is constantly interacting with the world, for He is the ground of the world's existence, if for even a moment God stopped sustaining the existence of the universe, the universe would just be annihilated and cease to exist.

As such, Allah, Exalted is He, is involved in history in that sense, but God does not enter the world Himself, not because He doesn't "want" to, but because it cannot happen. However, God does send us Prophets to instruct us to a path towards Him.

In this way God does act as a historical agent, it is He after all, who created the large flood, it is He who created the human race and give it His word, it is He who raised Jesus to himself.

Allah has ultimate control over everything, so he could be interacting without us knowing about it. But even if God doesn't directly interact, angels take their orders from God, so if angels do anything, they are doing what God wants them to do.

Historically, I think someone has mentioned Moses. There is also Noah and the flood where God has directly influenced history.

Yeah, you're right. God no longer involves himself in history, well, not anymore from my understanding.

Muslims only believe the Torah, the Psalms, and the Gospels, (if even those in reality). The Ten Commandments are not in the Qur'an, but the other events you mentioned are:

The plagues of Egypt: Sura 7.133-136
Noah: Sura 7.59-64
The parting of the Red Sea: Sura 26.64

Everything the Bible records in the books of history as God's work in history the Muslims do not recognize. So also God's acts in history to bring about man's salvation.

Muslims say that the God of the Bible is Allah. Christians would NEVER say that. As I mentioned, there are vast differences between them according to the Bible and the Qur'an:

YHWH is knowable (Jn. 17.3). Humans can come into a personal relationship with God. Allah is unknowable. He is so transcendent, so exalted, that no man can ever personally know Allah. He is distant, far off, and abstract. While Allah has personal qualities, no Muslim would refer to him as personal.

YHWH is a personal being with intellect, emotion, and will. Allah is not to be understood as a person. This would lower him to the level of man.

YHWH is a spirit (Jn. 4.24). Allah is not a spirit. That would demean him. He is neither a person nor a spirit.

YHWH is one God in three persons. The Qur'an explicitly denies the trinity. God is not a Father, and Jesus is not the Son of God. Neither is the Holy Spirit God.

YHWH is limited by his own immutable and unalterable nature. Thus God cannot do anything and everything. For instance, in Titus 1.2 we read that God cannot lie. In 2 Tim. 2.13 we read that God cannot act in a way that would contradict his nature. Allah is not limited by anything, not even by his own nature. He can do anything, any place, any time, with no limitations.

Because YHWH is limited, he is completely consistent and trustworthy. Because Allah is not limited, he is totally capricious and untrustworthy. He is not bound by his nature or his word.

The love of YHWH is his chief attribute (Jn. 3.16). The transcendence of Allah is his chief attribute. Neither does he have feelings towards man. That would reduce him to being a mere man.

YHWH enters history and acts to bring about man’s salvation. Allah does not personally enter into human history and act as a historical agent. He always deals with the world through his word, prophets, and angels. He does not personally come down to deal with man.

The Bible gives us both positive and negative attributes of YHWH (love, judgment, mercy, punishment, etc.). The Qur'an never tells us in a positive sense what Allah is like in terms of his nature or essence. The so-called 99 attributes of Allah are all negative in form, telling us what Allah is not, but never telling us what he is.

The grace of YHWH provides free salvation for man through a Savior who acts as an intercessor between God and man (1 Tim. 2.5). In the Qur'an there is no concept of the grace of Allah. There is no savior or intercessor.

They are both describing the same alleged god. Muslims believe that god sent plagues to Egypt, parted the red sea, gave Moses the ten commandments. It's the exact same sequence of events. He flooded the world but saved Noah and his family. Same event. You haven't provided any examples of anything that Christians believe God did historically that Muslims don't think happened. There is a difference in terms of their beliefs about the nature of Jesus but is that enough to say the difference is notable and vast? Whether Jesus was divine or not cannot be discerned from the stories about Jesus, which Muslims largely believe. So what is the vast difference?

It's one of the conundrums of Islam. They say that they recognize four books as inspired, but then they claim that three of those are so hopelessly corrupt that they become meaningless. It doesn't make much sense to me, but that's what Muslims teach. So they believe Genesis and Exodus, but they don't. They really only use the Torah to establish the legitimacy of Mohammed as a descendent of Ishmael, and therefore blessed by God.

The differences between Allah and YHWH are notable and vast. One of the chief attributes of Allah is his transcendence: he is not relational, but is distant and abstract. He does not act as a historical agent, but deals with the events of the world only through the Qur'an and his prophets.

Muslims literally believe in the same god that Christians believe was active throughout history. It's the same god that communicated with Moses, Abraham, Noah, and so on. They don't think the Torah we have is the same version god gave to Moses but all events that Christians think took place in history are events that Muslims think took place as well. I'm not aware of any exceptions to this. So if they think all the same events took place, wouldn't they have the exact same beliefs about god's interactions with humans throughout history?

> then what is different about their beliefs in terms of god interacting with humans?

Even though Muslims believe that the Torah was revealed by God, they also believe it is hopelessly corrupt and therefore unreliable. They believe that the Prophet Muhammad received his revelations from God through the angel Gabriel to correct human error that had made its way into the scriptures and belief systems of Judaism and Christianity (Suras 2.75, 79; 2.59; 4.46; 5.14, etc.). As such, they believe that only the Qur'an remains as a reliable holy text. Allah is only active in history in that he is active in the Qur'an, though which he has revealed himself, and he has sent 25 prophets, through which he has revealed himself.

> if Muslims believed that god had less interaction with humans throughout history it wouldn't provide any reasonable basis for thinking their beliefs are any more or less likely to be true than Christian beliefs

My point was addressing your question about Christianity being a historical religion, and Islam is not. What makes one religion's beliefs any more or less likely to be true are a consideration of all the factors of what each believes, not just this one slice.