To "cheat on someone" is a specific idiom in English. It means to be unfaithful, usually sexually, to a marriage or committed dating or romantic partner. It does not refer to cheating at cards, or cheating on one's taxes.
There was a pop song recently in which the singer complained that her ex-partner took up with soemoen else very quickly after they has split, and concludes that they new pair had been at least discussing getting together before the previous relationship ended. It includes the lines:
It took you two weeks
to go out and date her
so you didn't cheat
but you're still a traitor.
On general "cheat" means to violate rules, often for a financial advantage. It is frequently used for violating the rules of a game, or breaking a contract to gain a financial advantage.
Betray means "to prove unfaithful to, to violate an oath or promise". A spy is said to betray his or her native country. It implied a very serious promise and a very serious violation of that promise. One who betrays is a traitor.