"1 cup, drained" - does that mean measure before or after draining?

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.

I'd be interested to see the version of the pie you're making. I found a version on the America's Test Kitchen website that calls for candied yams in addition to the pumpkin and the recipe actually says

"1 cup drained candied yams from 15-ounce can"

So, if the rest of the recipe looks like it's the same recipe you're using, it looks like they may have actually reworded it to clear up the confusion you're running into.

It seems the answer is, drain them first, then measure them.

As a note, I think the place this issue is most important is when talking about flour... "two cups flour, sifted" is very different than "two cups sifted flour". With something like nuts (unless it's extremely central to the recipe) I'd argue that whether you chop first or second may not make that big of a difference in the end.

I'd guess that yams will fall into this second category, too. The small amount of difference between the two is unlikely to do much other than make your pie slightly less deep. In particular, since you're discarding the liquid, there's no reason to ever include it in your measurement. Because the yams are in pieces, it seems that all the liquid is going to do is fill in the gaps that the yams don't fill up. Regardless of whether you drain first or second, there shouldn't be a difference in the volume of the yams (assuming you don't pack them in, which isn't called for in either version of the recipe).