I know we're all different, but for me, there is no difference at all, between studio or live performances, I'm still giving a performance, the audience is just different, or just me!
A mate of mine, who enjoys going to gigs, says that later live performances are often better than the original record because the band have improved on it over time.
Now that could just be the live atmosphere that’s psychologically affecting his judgement or there might be something in it.
It might be worth delaying going into a studio with new tracks so that it’s second nature to play them and hone them first.
You hit a quite complex point. Most of my band's song change very much when played live - some because the original recordings have no drums or bass, others because I've had different melodic guitar parts on the way me and the backing sing it has changed over time, or we have added different harmonies and melodies. So in a way it's a good idea to wait.
In another way, however, it's hard to keep the raw excitement and novelty of the first few times you play a song after making it. With a band, there's that magic moment when, after messing about a fair amount of time, finally everyone "gets" their role and finally the song clicks.. that is imho the absolutely ideal moment to record it, even if if it's not perfect and certainly a performance is more polished after the 100th time you play it.