Chilling? How can I quantify that?
"Completely" definitely isn't a standardized term; however, since we're talking about making ice cream here, the correct answer is going to be "as cold as possible without freezing".

When making ice cream, it's important to get the mixture as cold as possible before you freeze it, especially if you're not using an ice cream machine, so that you give it as little time as possible for ice crystals to form.  If large ice crystals form, then you won't have ice cream, you'll have ice.

So get it all the way down to refrigerator temperature and work quickly when you take it back out of the fridge.  It'll be quicker and easier if you chill the bowl or receptacle first.