We returned Sun. from a weeks trip and bought our refillable popcorn bucket at Disney Springs. I was kinda bummed because it was an exact bucket of one I already have (same size etc.) My sister bought one the next day at AK and hers was different and shorter, but we were both able to refill the buckets every day with no issues at any park. I wasn't sure if we'd be able to refill after the 1st, since that's when it was supposed to end, but we did multiple times. For us, I'm not even sure being able to refill throughout the length of our trip really saved us any money. We ended up refilling it probably 4 times, but it was nice to try out.
There's also no doubt in my mind that we could have refilled ANY popcorn bucket. Like I said, the one we got from Disney Springs was a regular bucket (one we actually already have) and I asked a few different times when I bought it. I'm not sure if they just ran out, but either way I was never asked about it the different times I refilled it at different parks.