Common sense suggests that Elliott gave E.T. that name as nod to the fact that he was indeed an extraterrestrial. As you noted, the term 'Extra-Terrestrial' is in the title, and Elliott heard Steve use that term 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, E.T. is only ever referred to as E.T., never as Extraterrestrial or Extra-Terrestrial (the only time either of those words is spoken within the film itself is in the quote above). So it appears that E.T. is his full name (the one Elliott gave him, anyway) and not an initialism. Or to put it another way, the 'E.T.' part of the title is his name and the 'Extra-Terrestrial' is a noun that specifies what he is.