The camp will be split on this but I don't like the term warm up strokes. You are not loosening up, you are dialling in.
You are checking, double checking then triple checking your tip is lined at the cue ball properly and that your cue is pointing in the right direction.
I feel strongly that the practise strokes as you say should be only a few inches max. Very slow and deliberate and as close to the cue ball as possible. You get a better picture of where the tip is on the cue ball, more of the shift is in view meaning it makes it easier to align and use the cue as a guide for aiming. It makes it easier to keep the entire body still throughout until you pull the trigger which is a wide spread issue for all amateurs.