Do the Turing machines involved in Chruch-Turing thesis have to halt on all the inputs?
The Church-Turing thesis is an informal notion so it is not strange to see different authors taking slightly different positions.

To me, the second definition makes a bit more sense: CT tries to characterize "functions that are calculable in all cases", and for instance, we don't consider lets say, second order unification computable. The halting problem also falls in that category so I really wonder what do the authors mean.

Some interesting recent discussion about CT can be found in this thread: http://www.cs.nyu.edu/pipermail/fom/2016-August/020027.html