The official novelisation 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.”
The dialogue in the original script is slightly different from the film. Michael calls E.T. an "extra terrestrial" in front of Elliott. In Elliott's next scene, he's writing the initials "E.T." on his folder.
Scene 131.
[snip]Scene 136.
It may be of interest to you to learn that Spielberg and Mathison wrote a short treatment for a potential sequel, in which we learn that E.T.s actual name is Zrek.
In awe, everyone gazes upon the dark contours of the massive space machine. Suddenly the figure of Korel appears in an illuminated porthole. Telepathically Korel speaks to the children asking the whereabouts of the fugitive alien, Zrek. The children reply honestly that..."He's gone home!". Korel becomes angry, believing they are lying.
