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.

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.

Common would indicate that Elliott gave E.T. that name as nod to the fact that he was indeed an extraterrestrial. Elliott did use that term to refer to him earlier in the film.

TYLER: Hey, Elliott, where's your goblin?

MICHAEL: Shut up.

STEVE: Did he come back?

GREG: Wid he?

ELLIOT: Yeah, he came back. But he's not a goblin. He's a spaceman. As in extraterrestrial.

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.