Exactly. Peter turned down the FF because they weren't going to pay him.
I personally think that he turned Tony's offer down because he didn't want to be forced to hang with a bunch of old people. But when he was in space and saw an opportunity for major shenanigans (and adventure), well that was a more intriguing situation. At no point in Infinity War or Endgame did he ever ask to be an Avenger. Tony just made the spur-of-the-moment decision to "knight" him as one while they were flying to Titan. Far from an official induction, and then Tony died within like a day later, Peter time (if you take out the blip time).
In other words, he wasn't an Avenger in FFH, and he wasn't one in what came after either, and he still is not an Avenger.