There is a verb drinki which means 'to drink alcohol ceremonially or for pleasure' according to reta-vortato.de. So there will be difference between Ĉu vi volas trinki meaning water or tea and Ĉu vi volas drinki meaning beer or wine. How easily this could be used in communication? How popular is drinki?
Drinki is usually used in the meaning "to get drunk". I wouldn't use it, when you toast to something say in a gala.
You might want to take a look at a previous discussion "Trink-" versus "drink-"