Why is the korean war an example of a proxy war?

a north korea received military aid from the united states, while
china sent troops to south korea, even though the united states
and china weren't at war.

b. the soviet union sent troops to support the communist
government of north korea, even though the united states had
asked it not to

c. the united states supported south korea, while china backed
north korea, even though the united states and china didn't fight
each other directly

d. the united states supported a military coup that overthrew the
government in south korea, even though that government had
been democratically elected.