I do believe that cats purr for all of those reasons. My cats at home purr when they are happy and content.
But cats also purr when they are distressed. I work for a vet and have seen many cats purr when they are unhappy. We actually just had a cat come in whose owners slammed his tail into a door (he had to have the tip amputated). When we first saw him all he did was purr even though he was in pain.
No one really knows exactly why cats purr- that's one secret they will never tell us!