For the record, the "foreign snooker" is better referred to as "world snooker" (though I actually prefer to call it "English snooker" since that is in fact where the standards were set). Also, to confirm, yes you have the ball diameters exactly correct....2-1/8" is apparently the American standard (I've not played that myself) and 2-1/16" is the world standard (though with today's metric system, they actually define the ball diameter as 52.5 mm, close enough to not bother about the minute difference).
The actual size of proper world standard pockets is apparently a guarded secret as no public dimension actually exists. Instead, the size and shape of the professional pockets is very specifically defined as "conform(ing) to the templates owned and authorised by The World Professional Billiards and Snooker Association (WPBSA Ltd)." So there is not an actual number attached to it as when comparing to your 10 footer pockets, but the generally accepted number for comparison is that the corner pocket openings as measured at the fall is 3-5/16", slightly larger as you said than your 3-3/16" and coupled with the slightly smaller balls of the world standard. None of this is straightforward though because even restricting pocket openings to those exact numbers, any good table mechanic knows that pockets can certainly be made easier or more difficult in several ways ranging from how the rubber is shaped to where the "fall" actually begins in relation to the cushions.
All that aside, to the OP's point, I can't even imagine trying to play snooker with a pool cue. I don't play pool all that often so when I do, I am just using a crappy house cue, but I swear the thing feels like a huge, heavy log in my hands. I would probably put a two foot long tear in my snooker cloth trying (and failing) to apply proper backspin with that thing!