Let $M$ be a Riemannian manifold such that its isometry group $G=\textrm{Iso}(M)$ is a Lie group, and let $\Gamma$ be a subgroup of $G$.

1) What does the phrase "$\Gamma$ is a cocompact group of isometries of $M$" mean? Does it mean that the quotient

space$M/\Gamma$ is compact, or does it mean that thecoset space$G/\Gamma$ is compact ?

And

2) Is there an example as above in which $M/\Gamma$ is compact and $G/\Gamma$ is not, or viceversa ?

Also,

3) Is there any difference between "$\Gamma$ is a (discrete) cocompact group of isometries of $M$" and "The (discrete) group $\Gamma$ acts on $M$ cocompactly by isometries" and "The action $\Gamma\times M \rightarrow M$ is cocompact, where $\Gamma$ is a (discrete) group of isometries" ?