I am working on matrix Lie group actions and I want to figure out what we mean by maximal rank group acting on a manifold. Reading from "Representation Theory of Finite Groups" and "Finite group theory", I've understood that the rank of a group $G$ acting transitively on a set/manifold $M$ is the number of orbits of the double action $$ \sigma^2 : G \times (M\times M)\rightarrow M\times M,$$ $$\;\sigma^2(g,(m_1,m_2)):=(gm_1,gm_2). $$
On this paper https://www.ams.org/journals/proc/1972-035-02/S0002-9939-1972-0310808-1/S0002-9939-1972-0310808-1.pdf the authors talk about maximal rank action, do they mean that the group acts transitively on the whole product space $M\times M$?
My second "guess", looking at the proof in this paper, is that it is of maximal rank just if the differential of the map $$ \theta_m : G\rightarrow M $$ is surjective for any $m\in M$, i.e. $\theta_m$ is always a submersion. Is this the right definition?