You are correct. Progressive House is shortened today to Progressive. The House addition is a legacy thing, as it started with a change in House before even Trance was really a coined genre. Beatport uses the legacy name, though it fails to capture the actual genre it supposedly represents. Then again, Beatport genres are not always the most accurate things across the board. Progressive House is so ingrained into mainstream EDM that it saddens the amount of music out there I would miss since I no longer really hunt the genre (rather I hunt by labels I hear.)
You're also right about Progressive Rock and Progressive Jazz being genres... but neither really carried much weight or lasting power in their respective base genres.