Just as the term ‘space’ is used by some schools of algebraic topologists as a synonym for simplicial set, so ‘profinite space’ is sometimes used as meaning a ‘simplicial object in the category of compact and totally disconnected topological spaces’, i.e. in the other terminology a ‘simplicial profinite space’. This is further complicated by the question of whether or not pro(finite simplicial sets) and simplicial profinite spaces are the same thing.
The primary meaning (as Stone space) is used in sources on profinite groups, for which see the entry on such.
The second of the meanings is used by