"1 cup, drained" - does that mean measure before or after draining? I know this is similar to other questions like "1 cup chopped nuts" vs "1 cup nuts, chopped", but I wasn't sure about whether it applies to things that have liquid.

I was trying to make the pumpkin pie from America's Test Kitchen, and it calls for "1 cup of candied yams, drained".  If I follow the convention for nuts, then I would drain after I measure a cup.  But this seems imprecise, because there is a variable amount of liquid to begin with, so I would end up with a variable amount of yams afterwards.  It would be more precise to drain before I measure.

Does anyone have any thoughts on this?
Drain them first.  Your concerns are spot-on, and if you measured and then drained them, you would end up with less than 1 cup of yams.  Generally, you can tell be cause the recipe called for 1 cup of drained yams, not 1 cup of yams in their sauce.

Drain them, but double check that you won't need the liquid for anything later on in the recipe... That's a mistake I've made too many times with canned foods.