Just for accuracy...
The usual ship classifications are as follows:
Mainstream (known as "Contemporary" in the field): Carnival. Royal Caribbean, NCL. Costa , MSC
Premium: Holland America, Celebrity, Disney and usually Princess. I put Princess in this category because I truly believe they qualify in terms of food & service. Some travel agents disagree for the following reasons: many of their ships are getting pretty old, and some of the newer ones are as big as Carnival, but they do things like put five dining rooms on the ship so it feels more like an intimate ship. I actually prefer bigger ships because they have more options and activities - as long as they don't get crowded. I think Princess does a good job of crowd flow - as it is called. I have enjoyed every Princess cruise I have been on a great deal - some of my favorite cruises ever.
I have not taken an extended cruise on Celebrity except for a 1-week cruise in the caribbean on Solstice. It is a beautiful ship - bigger than Princess but holding fewer passengers.
I have been on short cruises on Millie and Summit. I do not find them that exciting, to be honest, too small for me for a Caribbean cruise. I do like small ships for port-intensive itineraries like Europe or Alaska, however, but if a ship has pod problems I would not book it for a destination-oriented cruise.
Deluxe: Oceania, Azamara and the small Princess ships have their own catregory: deluxe. Windstar is in this category usually as well. They are not quite luxury, but because of the open seating dining with alternative restaurants for a small passenger load they are considered close to luxury.
Luxury: Silversea, Seabourn, Regent and Crystal.