If a publication pays it contributors in anything other than copies of the publication or generally is only available by paid purchase, it would be professional. That's an OR, not an an AND.

So all of the examples you give are professional. They meet the "pay the contributors" criteria, even if they don't meet the "paid purchase" criteria. Only one need apply.

Actually, It think Tor.com probably is the functional equivalent of a professional magazine. It pays its contributors, although the content itself is available for free. Same with Baen.