Common sense suggests that the name 'E.T.' is indeed derived from the noun 'extraterrestrial.' As you noted, 'Extra-Terrestrial' is in the title, and Elliott heard Steve use the term 'extraterrestrial' before giving E.T. his name:
TYLER: Hey, Elliott, where's your goblin?
MICHAEL: Shut up.
STEVE: Did he come back?
GREG: Well, did he?
ELLIOT: Yeah, he came back. But he's not a goblin. He's a spaceman.
STEVE: As in extraterrestrial.
However, in both the film and the script, E.T. is only ever referred to by name as 'E.T.,' never as 'Extraterrestrial' or 'Extra-Terrestrial.' The only time either of those words is spoken within the film or the script is in the scene quoted above, where it was clearly being used as a noun rather than a name. So it would appear that E.T. is his full name (the one Elliott gave him, anyway) and not an initialism.