The [official novelisation][1] is a little clearer, and explicitly links the initialism with the words that it stands for.

> **“What am I gonna call you?”** Elliott looked into the great flashing eyes of the monster, where tiny blossoms of energy kept blooming and fading, to be replaced by others. The creature was sensing his way around, and Elliott stood back to give him room. **“You're an extraterrestrial, right?”**
> 
> [snip]
> 
> “You keep that towel,” said Elliott. “That's yours. **We'll mark it E.T., for extraterrestrial.”** He touched the monster again, amazed by the texture of the skin. Another wave went through Elliott and he knew that the creature was older than Methuselah, older than old. “You're something like a snake too, aren't you. Boy, you are really weird.”


  [1]: https://ettheextraterrestrial.fandom.com/wiki/1982_novelization