Technically no, you are correct. But realistically, one can totally reasonably try to make a claim that Elon was a founding member. The company was created in June 2003, and in February 2004 Elon already became the largest shareholder (which is just slightly over half a year later after the company was founded). And that's 5 years before they produced their first ever car (the original Roadster) and 7 years before their first mass produced car (Model S).

Though I am skeptical of the sourcing on that quote, given that Elon isn't listed as a founder neither on the official "origin story" page of Tesla nor in the wikipedia entry for the company.

Correct. His first involvement was as an investor in their series A. He does, however, say[1] he's a co-founder, so I guess the quote is technically accurate.

[1] https://www.tesla.com/elon-musk

He didn't, he just pretends he did. Same as how he LARPs as an engineer for SpaceX.