Do the Turing machines involved in Chruch-Turing thesis have to halt on all the inputs? The following is Church-Turing Thesis from two books.

Is it correct that


The first book seems to say that the Turing machines involved in the thesis may or may not halt on a given input,
the second book says that the Turing machines involved in the thesis must halt on all the inputs?


Do the two books contradict each other? Why?

Thanks.

From Ullman and Hopcroft's Introduction to Automata Theory, Language, and Computation 1ed 1979


  Note that one TM may compute a function of one argument, a different
  function of two arguments, and so on. Also note that if TM M computes
  function f of k arguments, then f need not have a value for all
  different k-tuples of integers. ...
  
  In a sense, 
  
  
  the partial recursive functions are analogous to the r.e. languages, since they are computed by Turing machines that may or may not halt on
  a given input. 
  The total recursive functions correspond to the recursive languages, since they are computed by TM's that always halt.
  
  
  ...
  
  The assumption that the intuitive notion of "computable function"
  can be identified with the class of partial recursive functions is
  known as Church's hypothesis or the Church-Turing thesis.


From Lewis and Paradimitriou's Elements of The Theory of Computation


  However, we have also seen in the last chapter that not all Turing
  ma­ machines deserve to be called "algorithms:" We argued that Turing
  machines that semidecide languages, and thus reject by never halting,
  are not useful computational devices, whereas Turing machines that
  decide languages and compute functions (and therefore halt at all
  inputs) are. Our notion of an algorithm must exclude Turing machines
  that may not halt on some inputs.
  
  We therefore propose to adopt the Turing machine that halts on all inputs as the precise formal notion corresponding to the intuitive
  notion of an  "algorithm".  Nothing will be considered an algorithm
  if it cannot be rendered as a Turing machine that is guaranteed to
  halt on all inputs, and all such machines will be rightfully called
  algorithms. This principle is known as the Church­-Turing thesis.
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