It’s spelled Ovechkin, and he’s Russian, not Thai.
If you and your companions are paying separately you’d ask for “separate checks”. I’ve never heard anyone in an American restaurant ask for “separate bills”, and I think it would sound really weird.
You could “split the bill” or “split the check” pretty much interchangeably, though.
And if you’re going over the list of charges to confirm everything’s been priced and added up correctly, you’re never “checking the check”, you’re “checking the bill”.
Or “owerkin,” as there is no V sound in Thai. 
(It’s popular for Thais to spell their names in English with a V, but it’s always pronounced as W.)
Unless you’re in Prague, in which case you’re “billing the Czech.”
And if you order a nice German pale lager but it’s not sufficiently cold and you ask the waiter to bring an ice bucket so you can cool it down, you’re… oh, never mind.
Huh?
I usually say check but sometimes I ask for the bill. Check seems slightly wrong to me, like I’m asking to be paid, right now, for eating there.
Check. Long Island New York. A bill is something that comes in the mail.
“Blowjob, please. Wait, not blowjob…check. Yes, that’s it. Check, please. Whoops.”
“To pay the scot” is an idiom meaning to pay your reckoning, your tab, your accumulated charges. Apparently originally derived from a particular type of taxation.
Check.
Bill makes it sound like a phone bill or water bill.
Is that anything like “paying the piper”?
Always heard “Check”. I’ve yet to hear “Bill” when referring to the restaurant tab.
A duck walks into a midwestern U.S. pharmacy:
“Give me some chapstick and put it on my bill.”