A mashhūrah (commonly accepted premise) is accepted within a community or discipline. It need not have been expressly conceded by every opponent.
A musallamah (conceded premise) is admitted by the particular disputant. The admission may be limited to that exchange and may not be widely known.
For example, “the grammatical subject is nominative” is a recognized rule within grammar. A special condition admitted by one opponent can be a conceded premise in that discussion even if it has no public standing.