TL/DR: we are pretty much all "home players" as long as we practice on a regular basis. Some of us play in other environments as well.
You're trying to make an obvious meaning far too specific. FYI, the phrase in the subject line is "bedroom player," which carries the implication that that's the only kind of place an individual plays. The word "bedroom" could be replaced by "living room," "garage," or "man cave" with no change in meaning. OTOH, the phrase I used - "gigging musician" - does not mean only people whose sole income is from music performance. I've known and worked with quite a few working musicians who made substantial portions of their income from nonmusical activities. I've been in that group myself at times. We were nonetheless working musicians. My definition of "working musician" means that you derive at least some of your income from musical activity, including performing live, recording, writing, arranging, and/or producing.
FYI, the distinction carries no stigma re:skill level. I've known some killer players who only ever play at home.