A group action is defined to be action of a group $G$ on a set X.
But a group representation is also defined to be the action of the group $G$ on a vector space $V$.
So in fact both the definition are same except some less condition on group action.
i.e., any representation is a group action but the opposite is not true.
Is it like that?
Yes, it is "like that". Every representation on a vector space is an action on the underlying set. There is an extra condition: the permutation of vectors corresponding to each group element must be a linear transformation.
That extra condition makes no sense when the set $X$ acted on is just a set with no other structure. When it does have structure (perhaps it's a manifold) we usually require the action to respect that structure.