I'm not sure how to answer your question because I usually plant them in potting soil mix, not water.
I cut onions saving the bottoms in a pyramid shape. Larger onion bottoms get the pointy tip cut off. Most of the time, I leave them on the counter until the center turns green and push up. The I *remember* to plant them.
Sometimes, it's the last bit of onion that had been forgotten in the fridge, in which case, the roots start to grow while it's in the plastic wrap/zip bag.
I generally snuggle the bottom 1/2~2/3 in the soil with any handy container plant in the kitchen, but I have occasionally planted them by them selves -- about 2 to a 4" pot, 3 to a 6" pot.