TM1 accepts w1 vs TM1 halts on w1
Your intuition is correct in defining the two problems. In fact accepting also means halting but halting does not necessarily mean accepting (it could also reject).
Remember that when a Turing Machine is run on an input it can do three things: Accept, Reject, Loop Forever. 
In the first two cases the machine stops, in the third obviously not.

As for decidability, note that both problems are the Halting Problem because the first case  implies a halting, even if it excludes the fact that 
the arrest is due to reject, therefore they are both undecidable.