Don't rinse here at all. If there is a wad I can pick up with up with a baby wipe I do my best and toss that in the garbage - not taking the risk of plugging the toilet as by change time it is now a ball/chunk and not a nice flushable snake.
The more parents get the little bags of stuff home - and I even ask them for some bags although I do keep a box of cheap zip bags on hand to seal in the mess when necessary.
There is no way having an open pail of water is safe in daycare and especially if it needs to have bleach in it. I had a mom using cloth diapers offer to bring me a pail to put water and soap in and then she was going to take it at the end of the day and I said no way. Even with those the diaper and any contents not able to be picked up with the wipe is simply put in the bags they provide.
I do have access to my laundry tub and have rinsed a puked on tshirt or something in there with the idea of whatever the running water washed off fine. And then during naptime did a bleach and water thing in the sink. Since this is a less than 5 times a year thing I have done it. For potty mistakes nope goes home. If child is having that many then maybe they need more time in a diaper while still being taken to the toilet regularly.

