It means you're a full-time professional musician. Not a part-time semi-professional musician with a day job to pay the bills, who is permanently distracted by Not Music.
>a lot of independent musicians who have achieved massive success outside of the traditional recording studio model in the internet age, largely through selling their own work and through touring.
No there aren't. There are very, very few professional musicians working that route, and hardly any of them have achieved 'massive success.' Most of the ones who did already had an old-style label-based career, or at least label+management backing.
Most of the really successful ones don't sell their own work and perform. They write and arrange songs for top performers, or they work as producers - which is what Albini does now, mostly - or they do work-for-hire movie and game soundtracks, or they write commercial music for ads, or they work as sound designers.
Being a famous name performer is a career that works for a tiny handful of (mostly) singers, who are primarily picked for sexiness and (sometimes) charisma, not for musical creativity.