8" vs 10" ???
The rim should be based on the tire tread width, not the 12.5 but the actual tread width. The easy way is to take what the manufacturer says not the supplier.
The narrower the rim the narrower the actual contact of the tread. If you air down some say it is better to have a wider rim because the tire tries to spread itself out.
The 8" rim will also give you a little more rim protection.
Watch the backspacing, it should be the same on both rims. If you have the same backspacing then the tire will stick out more with the 10" rim and in turn cause more interference with body parts, as Perry stated.
As usual the best thing to do is look at another vehicle with the same size tires you plan to run.
If possible the vehicle should be doing the same type of trail riding, and see how the tire look, if it is what you want then you have your answer.
Most selections on any vehicle are personal preference.
I use to run an 8" rim on 10.50s w/9.25" tread width.
My new tire, coming soon, will have a tread width of 12.50" and I plan on running a 10" rim. (17x13.5x16 overall tire size)
Good Luck on the selection, make it the way you want it...