What is a word for getting an award in a competition or being one of the top three?
You're perfectly right, we use the general term "place".
I was the first in my school to place in the annual spelling bee.
You could also use "award" without specifying which.
I won an award at the annual poker competition.
For the Olympics, since they award medals, they use "medaled".
He medaled in three events at the 2000 Olympics.