Back from my trip to Baku, so I can answer the question myself. The Excelsior is not bad at all, but lacks the "local meeting place" character of the Hyatt and has less attractive public areas. The Excelsior was available at an entry rate of US$ 300, whereas the Hyatt cost at least US$ 400.

I really liked the Park Hyatt Baku. Fantastic service (especially considering that this is the old Soviet Union), top fitness center, tasteful lobby that still allows smoking cigars (sorry to the non-smokers!), very good rooms. Only downside is the slightly off-center location and the crowded main pool (better go to Hyatt Regency pool next door).

I would say that the Hyatt is a luxury hotel, even at the demanding expectations of this forum.