I guess I am somewhat of a minimalist when it comes to notation. I have spent a lot of my career writing about group actions, and what I usually have done is start out defining what a group action is, and then say something like If we have in mind a fixed action of G on X then we will say that X is a G-space and we will denote by gx the result of the action of g in G on an element x of X.''
To be more pedantic, the point is that for a fixed group G, G-spaces form a category and if you ever had two different actions of G on the same topological space, you really should use two different symbols to denote that space with the different actions And it is convenient to have a morphism (equivariant map) $\phi$ between G-spaces just mean that $\phi(gx) = g \phi(x)$.