I've been reading Valette's introduction to the Baum-Connes conjecture and as I read the example of a construction of a (model of the) classifying space for proper actions of $\Gamma$ (discrete) given an affine, isometric action on a Hilbert space $H$ that is "proper" I came to realize that there might be some inconsistent definitions for the latter word.
The definition given one/two pages before by Valette would say that for $v,w\in H$ there are balls $B_1$ and $B_2$ around $v$ and $w$ such that $\{\gamma:\ \gamma B_1\cap B_2\neq \emptyset\}$ is finite.
The definition that is implied in the example (when proving that the construction actually yields a paracompact proper $\Gamma$-space) is that for all bounded $B_1$ and $B_2$ in $H$ the set of elements $\gamma\in\Gamma$ such that $\gamma B_1$ meets $B_2$ is finite. This second terminology is consistent with that used for proper cocycles (there's a one-to-one correspondence between these and affine isometric actions) on Bekka et al's book on Property T.
My question is then: what is, if any, the correct definition? Or are both terminologies acceptable and one must deduce the meaning from context?
EDIT: after some Googling I came to the conclusion that the last option is the correct one.