Perhaps it would be helpful to trace through a few instances of Jesus's use of Ἐγώ εἰμι (ego ay-me: "I am"). We will see that the context gives us the intended meaning.
[list][*] John 4.26. Jesus is identifying Himself as the Messiah about whom the woman made a statement. Leon Morris writes, "His use of the emphatic pronoun in this expression is in the style of deity. 'I that speak to you, I am.' "
[*] John 6.35: "I am the Bread of Life." Jesus is using the Ἐγώ εἰμι to depict His identity and a function: the giver of life (v. 51). As has been said, He uses this same form as an emphatic statement of His identity "with overtones of divinity" (Jn. 6.41, 48, 51; 8.12; 10.7, 9, 11, 14; 11.25; 14.6; 15.1, 5). His self-identification as the Bread that comes down from Heaven is a distinct allusion to the manna in the wilderness.[list]
If we look at 8.24, translators use "I am He" because they consider that it flows more naturally in the English. Jesus's phrase is, "ἐὰν γὰρ μὴ πιστεύσητε ὅτι ἐγώ εἰμι" ("for if you don not believe that I am"). It would work just as well to read it that way: "I told you that you would die in your sins; if you don't believe that I am (meaning His existence and identity as the Messiah from Heaven [ the preceding context]), you will certainly die in your sins."
In 8.58, to translate Jesus's statement as "I am he" doesn't make sense. Jesus is claiming pre-existence to Abraham, so to be declaring Himself as eternally existent "Before Abraham was, I am") is a more accurate translation than to say, "Before Abraham was, I am he." That translation doesn't take us anywhere or clarify what Jesus is claiming. In v. 59, they picked up stones to stone Him for alleged blasphemy.
Stephen Homcy writes,
"The sacred name of the God of Israel underlies Jesus’s claim, and He is clearly asserting His preexistence. The contrast between the verbs genesthai (aorist) as applied to Abraham and eimi (present) as applied to Christ is striking. C.H. Dodd says, 'The implication is that Jesus does not stand within the temporal series of great men, beginning with Abraham and continuing through the succession of the prophets, so as to be compared with them. His claim is not that He is the greatest of the prophets, or even greater than Abraham himself. He belongs to a different order of being. The verb genesthai is not applicable to the Son of God at all. He stands outside the range of temporal relations.' "