A couple of issues:
1. The piling contractor should cut the tops to the required elevation. They are his piles and you do not want to get into further liability of potentially damaging his work by cutting them yourself. This should not even be an issue on your part...not your job.
2. If piles were not anticipated in your bid, and you had nothing other than boring logs and insufficiently detailed plans, you should have a valid claim for extra pay for the more tedious and time consuming excavation around piles...again you have to be more careful not to damage the pile contractor's work.
As for the mat/raft designation, I've heard them throughout my career to be interchangeable; however, BigH's statements make sense...I'm just not sure that most use the vernacular in that manner. As for pile support, it generally is stated if they are pile supported (piled mat, piled raft, pile-supported mat, etc.).
Is there anything indicative in the design of the mat reinforcing that would lead you to conclude that there is a pile interface?