I'm not an expert, but I'll tell you what works for me.
Big picture: yes, you want the gun to be absolutely still when you fire. You are at the bench to get the Point of Aim and Point of Impact together. You want to eliminate as many variables as possible. When you've found out what the rifle/load/scope combo will do, then you move to phase II, which is shooting under realistic positions/scenarios etc. to find what YOU can do with that rifle.
First, I like to be sitting more upright, so if the bench is low, use the ammo can/lots of sandbags/purpose-built gun rest, etc. to get to a comfortable position. If you are hunched over, you are more unstable. And the recoil will be moving your shoulder parallel to your spine. It does not like to move that way and recoil will feel a lot worse. You'll be more likely to flinch.
If you sit upright, recoil moves your shoulder perpendicular to your spine. You get much better shock absorption that way. You'll bemore comfortable and you'll get better groups.
Second, you want to arrange the sandbags/gun rest etc. so the crosshairs are as close as possible to where you want them. You want to direct the rifle as little as possible. Let the rest do as much as possible. You want to prevent your heartbeat, breathing, pulse etc. from affecting the shot.
The best situation is if the rifle is abslutely motionless and on-target when the trigger breaks. If you can do that regularly, you'll know what that rifle/ammo/scope combination can do.
Once you've got the sights adjusted and your load selected (remember, for best results you should sight-in for a particular load; if you change loads, re-adjust sights), then start shooting in the positions and settings you will be hunting in, insofar as possible. Go plinking. That will tell you how well you can shoot that particular rifle/load/scope combo. The goal is for you to get as good as the rifle--that is, to minimize the difference between bench-rest groups and field groups.
That's my amateur story. Now I'm as eager as you are to hear from the real experts.