In answer to your question, a stay is the time from check in to check out. But a little twist is if you check in spend the night and check out then check back into the same hotel the same day you checked out you do not get an extra stay out of it just more nights. Must change hotels to get more stays on consecutive nights.
Some people say it is easier to qualify on stays. I have never qualified on stays, only nights and have been diamond going on my 4th year.
Hope this helps your understanding of what a stay is.