Does this have to be a song that was ONLY written that someone else performed? Like it was written but never performed/recorded by the composer?
No. As long as the person established his/her reputation as a writer first. Many many of these individuals also went on to record their earlier songs too.
I think Willie Nelson had recorded both songs I put above, in 1961, but he was pretty much a nobody at the time.
See "Hugh" Nelson reference per that NPR podcast (now "Willie"). Also the informational YouTube of him and Faron Young I think is mostly them joking at how he wanted Faron Young to buy the song for $500, Young refused, and Nelson made something like $20,000 that first year in royalties (IIRC). Faron Young was a big star then. Per NPR, they also didn't know at the time Patsy Cline's "Crazy" was going to be a big hit.
Furthermore, that 1962 album by Willie is about songs he wrote, consistent with the name of the album. It is also consistent with that introduction in 1965 at the Grand Ole Oprey. Not sure, but I would guess that was his first performance there (possibly televised, as there is footage of it).
Side note, Patsy Cline had died already in a plane crash (1963) by the time of Nelson's Grand Ole Oprey performance, if the 1965 note in the YouTube video is correct. I think a different YouTube of it had 1962 in the YT title, but that is when that album was released, so I'm guessing the 1965 is correct.
Another side note, the Wikipedia about him performing Whiskey River on the very first "pilot" episode of Austin City Limits mentioned how he hated performing on TV-- but turned out he liked the more informal aspect there. I guess they liked the pilot, and Austin City Limits began production as a regular program a year or two later.