OP, good post...
Life is never about titles, basically you can be what you aspire to be. Pretty cosmic, eh? What I'm hearing is that the term "musician" to you translates to acceptance with your peers (at whatever level they are). Music is similar to public speaking, to do it well requires two things: first knowing your content, and second confidence. Having a great speaking voice and having charisma really help, but I compare that to talent and experience.
I know that for myself, I aspire to play beyond my abilities - and that drives me to practice and improve. I took a 10+ year hiatus when a band thing didn't work out in the late 80's and really didn't resume playing until around 2001 (in hindsight it was better that I stuck with my day job

). Joining a small private club band for fun got me back into playing, I stopped playing as a guitarist and returned as a beginning bassist. Now almost 20 years later I play with group of talented "musicians" that push me to actually put in the time to "earn" the chops I've been able to "ad-lib" over the years (also self taught).
So my 2c's are cut yourself some slack and enjoy playing, sounds like you're working to get better. I'm more comfortable playing with my band, where I've practiced the songs and know the set. But I still do the occasional jam, or open mic where you are forced out of that comfort zone. Like public speaking having a good understanding of the subject will build your confidence. In bass terms, knowing your basics so you can adapt to whatever the music is. On those times I get lost, I just turn down a bit... at least it's not so obvious.