The terminology is unfortunately confusing as these properties are in a sense "orthogonal". The definition in terms of existence of an invariant mean on the action space (your definition 2) goes back to von Neumann, whereas your definition 1 goes back to Zimmer who introduced it (in the measure category and in a somewhat different form though) in 1977. Amenability of the group $G$ is equivalent to the amenability in the sense of Definition 1 of its trivial action on the one-point space and to the amenability in the sense of Definition 2 of its action on itself.
For an explicit counterexample to your claim take the boundary action of a free group. It is topologically amenable in the sense of Definition 1, but not amenable in the sense of Definition 2.
A counterexample in the opposite direction is provided just by the trivial action of any non-amenable group on a singleton.