Elmo has had his name as early as season 12. A season 12 press release described Elmo as a character who talks in sounds (though obviously there are early segments where he speaks full sentences, perhaps from before he became Elmo). Brian Meehl has an official website, and I e-mailed him shortly after seeing that press release, and he told me that the early Elmo often spoke in single-words and was learning to speak spanish (the fact that he was learning spanish was mentioned in the 40th anniversary book). It was said in Street Gang that Brian Meehl actually only performed Elmo four or five times, though he seems to be considered Elmo's original performer (there are known instances where Elmo was performed by others, before Meehl took on the role). Meehl told me that he only remembered one speciffic scene where he performed Elmo.
In Caroll Spinney's autobiography, he said that the early Elmo spoke fast and often repeated words. In fact in Follow That Bird, Elmo seems to be doing this, saying "No, no, no, no, no!" Hmmm, that was also part of his line in "Me, Claudias". Spinney gave one example of his early speech being "Play, play, Elmo want to play". I guess that implies he spoke in the first person early on, though it could just be an example of how he spoke.
Fans have said that in the original "We are Monsters", before the monsters sing Elmo originally said "He doesn't look like me", which was redubbed to "He doesn't look like Elmo". Somebody at Muppet Wiki contacted someone at Sesame Workshop who confirmed that the pre-Kevin Clash version debuted in season 11 or 12, but I don't know if anybody actually has a video copy of the original edit. Every version I see on YouTube seems to take the video footage and replace the audio with the version from The Gang's All Here (where all of the audio tracks originated on the show as opposed to being rerecorded), and I've noticed when Elmo says his opening line, his mouth makes the same number of syllables as "He doesn't look like Elmo" (instead of "Me"). So maybe Elmo did speak in the first person at first.
If more classic episodes show up at iTunes, I hope they include a season 11-17 episode with the song. Or maybe sesamestreet.org will add it (I never thought a green Grover or Brian Meehl as Telly clip would be added to the site but the site does have those kinds of clips).