I use a scalpel (#3) and a fresh sharp blade (#11), both Swann-Morton numbers. I run the blade in lightish strokes in the two long "furrows" then the short furrows at each end. The long strip of counters should be almost detached by now. The only place where you might need moderate force is at the very corners when finalising the ends of your cuts where they join. You will now have a strip of 10 or 2 x 10 1/2" or 8 or 2 x 8 5/8" counters. If a 2 x 10/8 strip, then again a light run or two with scalpel along their long joining furrow will give you 2 strips of 10/8. For each strip I then detach each counter by gently positioning the blade into the gap so that the blade edge is horizontal and when happy with my placement do a single chop.
Providing you keep the blade vertical or fractionally canted towards surround (aka the tree) there should be no risk of shaving off a strip of the printed part of a counter, only the surround.
The most important bit is having a very sharp blade. This means that you can cut well with much less pressure. While you might need 2 or 3 strokes, light pressure means greater control over the scalpel+blade and less likelihood of going astray or even snapping the blade, impaling your fingers and getting blood on your counters. Using light strokes delays fatigue and hand shake.
While I use a modeller's cutting mat, any piece of wood that has a flat surface will do. Cutting directly on the kitchen table may ensure domestic combat, so use something like a cutting mat or disposable piece of wood or plastic (never metal, that will blunt your blade in no time).
One additional suggestion is have enough of a gap between your surface and your head and shoulders so you can look vertically down at what you are doing. That has two advantages in that little is masked by scalpel and blade and you have your weight to assist. Even though you will be using light pressure, the less your muscles have to do the longer your hands stay steady. So maybe put a cushion on your chair or put your cutting mat on a low table like a coffee table. Once you go beyond a couple of counter sheets, muscle fatigue is the enemy of accuracy.
So in summary, very sharp blade, (multiple) low pressure passes, appropriate cutting surface and a comfortable stance/posture.