A single factor isn't helpful—the test for determining the "nature" of a publisher has two factors.
- Does the author have legal title to the books as they come off the press? If yes, it is probably self-publishing. If no, it is either commercial or vanity publishing.
- Does guaranteed capital flow as the first copy comes off the press point toward or away from the author? If toward, it is commercial publishing. If away, it is either self- or vanity publishing.
N.B. "Guaranteed capital flow" means accounting for
all required capital expenditures necessary to publish a book. That includes mundane things like copyright registration, contract drafting and review, and so on, in addition to obvious things like advances (and, conversely, up-front or back-end fees and/or required purchase of copies by the author).