The answer is... it depends...
Multiple internships- a lot depends on which theatres they are. You want to make sure that multiple internships don’t become lateral moves. If you are doing more than one, make sure each one is better than the one before whether its because it is at a bigger larger regional theatre that is well known or because you get to PSM the children’s show while working on the main stage or you'll be working with a stage manager with some knowledge and connections that you can benefit from.
Definitions are all different. What I did as an apprentice at Round House Theatre is the same thing that Dramachic did as an intern at Olney Theatre Center. I believe that the exact same job is a fellowship at Arena Stage. Round House's interns are non-paid HS kids and at Olney the apprentice works with the education department. No matter what it is called, check out the job requirements and the pay/stipend/benefits. Luckily most theatre employers understand and will ask if there is any question as to what your duties were as a fellow/intern/apprentice.