I think there is an issue with the definitions for each of the words you've provided.
To bury a person, you would first have to lay them in the hole and then cover them in soil.
It's worth noting though, that you don't always need to lay something in a hole to bury it. You can just pile stuff on top of anything to bury it.
Lay - to put something in especially a flat or horizontal position, usually carefully or for a particular purpose Cambridge Dictionary Def.
Bury - to cover something or someone completely with a large quantity of something Cambridge Dictionary Def.
Using your examples:
They laid him in the old churchyard.
They put him [down] in the old churchyard. (Note that this isn't necessarily in a grave)
They buried him in the old churchyard.
They covered him in something in the churchyard. (Note that you would always assume that if you bury someone in a churchyard, you always mean that you have buried them in a grave with soil.)