What does E.T. mean in E.T. the Extra-Terrestrial?

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.}}

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.

Is "E.T. simply an initialism of "Extra-Terrestrial"

Yes, it is.

The novel "2001: A Space Odyssey", written by Arthur C. Clarke was based on the film screenplay written by Clarke and Stanley Kubrick. The novel was was published in 1968, and Chapter 32 is titled Concerning E.T.'s.

The chapter begins with a discussion of how scientists were debating faster-than-light travel, then continues with:

If there was disputation among the physicists, it was nothing compared with that among the biologists, when they discussed the hoary old problem: "What would intelligent extraterrestrials look like?"

[skipping a few paragraphs]

Of course, there would be minor differences like six fingers instead of five, oddly colored skin or hair, and peculiar facial arrangements; but most intelligent extraterrestrials -- usually abbreviated to E.T.'s -- would be so similar to Man that they might not be glanced at twice in poor lighting, or from a distance.

Clarke, who had good science credentials as well as writing credentials, explicitly said "E.T." == "extraterrestrial" almost 20 years before the release of Spielberg's film.

Added after a comment saying "that doesn't show a link to the names Spielberg used in his film":

Spielberg cites this film as highly influential, along with other Kubrick films and other directors (quoted here and here). Although the 2001 film doesn't use the term "extraterrestrial" or its abbreviation, the popularity of the film boosted the popularity of the novel as well.

I can't find a source where Spielberg says he read the novel, but he does talk about making up an extraterrestrial as an imaginary friend as a teenager, several years before the 2001 film and novel were released. For me it's very likely that between 1968 and 1982 Spielberg read the 2001 novel or other science literature that linked the technical word with its abbreviation.