counter for cupfuls, wine glass, glass, toast
杯 centers on a vessel for drinking: a glass, wine glass, or cup, and by extension a toast or a counter for cupfuls.
杯 combines 木 (wood) with 不, which likely contributes the sound. The character originally referred to a wooden drinking vessel.
A wooden 木 cup that is not 不 full—just a glass for a toast.
For ハイ, imagine raising a glass and saying 'Hi!' as a toast: Hi -> ハイ.
one cup (of); one glass (of); one bowl (of); cupful; glassful; bowlful; spoonful
sake cup; cup for alcoholic beverages
cheers; bottoms-up; prosit
the best one can do; one's best effort
full
having one's hands full; not having any room to do more; at the limit
toast; congratulatory cup
gold cup; gilded cup; goblet
Emperor's trophy
World Cup
let's have a quick drink
skull cup
sake cup shaped so it cannot be put down until emptied
Asian Cup; Asia Cup
Daily Cup (horse race)
Davis Cup
to have a drink; to have a few drinks
to the limit; to the fullest extent; absolutely the limit; the best one can do; breaking even
to have a drink; to grab a drink
tipsy; a little drunk; slightly intoxicated