It's interesting indeed that Turing's machines as he defined them can never halt:
> he does not discuss the halting of his machines at all,
and makes no provision for the computational processes undertaken by his machines
ever to stop; in particular, he has no convention as in contemporary accounts of
a halt state for the machines
So instead of asking whether they halt, Turing asked whether they ever print a particular symbol. Of course one could call that symbol the halting symbol, and adopt the convention that printing the halting symbol amounts to halting. So while Turing did not name it the "Halting Problem" he proved an obviously equivalent result.
> he does not discuss the halting of his machines at all, and makes no provision for the computational processes undertaken by his machines ever to stop; in particular, he has no convention as in contemporary accounts of a halt state for the machines
So instead of asking whether they halt, Turing asked whether they ever print a particular symbol. Of course one could call that symbol the halting symbol, and adopt the convention that printing the halting symbol amounts to halting. So while Turing did not name it the "Halting Problem" he proved an obviously equivalent result.