What is the proper/most common way to serve curry?
There are two traditional way to eat curry. Indians typically eat their curry with a type of bread. Usually Naan or Roti and use piece of the bread as a scoop/spoon of sort. 

Thai curry will tend to be eaten, as you suggested, over top of rice. Typically it is separated when it is served so the rice does not get soggy while it is waiting to be served. The rice is typically served on a plate big enough to fit both the rice and the curry.

Of course, you can eat the rice and curry however you see fit. 

Also there are thin soups that is flavoured with curry but usually those aren't served with rice so there shouldn't be confusion in that category. But do note that if you think the curry is too thin, it will thicken as it mix with the starch from the rice.