You showed a reduction from -(Htm) to Htm/all (that is, from the complement of Htm to a class of M's that stop for every input).

f:<M,w> -> <M'>

M doesn't stop on w iff M' stops for every input

You did it by restricting the number of steps to |x| (x is the input of M'):

if M stopped: M' will enter a loop

if M didn't stop in |x| steps: M' will accept x

But I don't get the reasoning:

We were supposed to show an "iff" connection to <M doesn't stop on w>, and not to <M doesn't stop on w in |x| steps>.

Weren't we supposed to show that the class {M | M doesn't stop in k steps} actually belongs to coRE first ?