The "first class" or luxury or whatever else term might be is not based on not allowing children.
Viking along with Oceania, Azamara, Windstar, Paul Gauguin (that became part of Ponant) is upscale or premium, not luxury. We've been on Azamara and Paul Gauguin.
Besides Silversea and Seabourn, luxury are also Regent, Seadream, Ritz-Carlton Yacht Collection, Scenic, Ponant, Crystal. We've been on Regent, Ponant, and Seabourn.
A Cut Above: The Difference Between Premium and Luxury Cruises - Luxury Travel Magazine