I'm the family cruise editor here. Actually Princess used to take 2 year olds, but don't any longer. Their youth program starts at 3 years old and kids must be potty trained.
For two year olds, check out Norwegian Cruise Line and Carnival, since both accept kids two and over in the youth program. Kids don't have to be potty trained to participate. Carnival changes diapers while NCL will give you a beeper so you can come change a dirty diaper. Both have good youth programs.
Feel free to ask any add'l Q's.
Luisa Frey Gaynor