I am a firm believer in mounting a scope as close to and as parallel to the bore as possible. That part is easy. The other issue then becomes the dimensions of your butt stock as well as the dimensions of your face.
You did not say what distances you will be shooting at and if you are going to use 0 MOA bases (or Rail) or bases (rail) with some MOA Built in. This will make a difference.
Draw yourself and exaggerated picture of the line of convergence between the scope line and the bore line at different mounting heights above the bore and you will see the difference. With a scope sight line above the bore, you are going to be on target at two yardages, one on the bullet flight upward as it crosses the line of sight with the scope and one as it is coming back down through the scopes line of sight.
So, you set up a set of rings that allow you to have the scope on the gun with the objective as close to the barrel as possible and you might have some other issues, like eye relief adjustment as the barrel is tapered and viewing based on either the dimensions of the stock or your face.
I only use one brand of Scope Rings and I have multiple sets in both different diameters and different heights. They are all pre-lapped. For a new rifle or new scope / rifle combo I play around to I get exactly what I want.
With regard to the Medium Verses High etc. those measurements vary by manufacturer so a High from one manufacturer is not necessarily the same height as a High from another. Look up the exact height dimensions on the various manufacture web sites and you can confirm that.