I think this should be called the relatively ample cone and its closure the relatively nef cone. Yes, this already exists and is sensible.
This dissertation might be somewhat useful (although things are done for smooth $X$ instead of singular $X$).
EDIT: With regards to your two conditions (and the comments above), I think this condition is basically factoriality for the singularity. Suppose $x \in X$ is an isolated singularity as you describe and suppose that we are working locally (in the affine setting) with only one singular point.
Suppose that $X$ is factorial, this means that every divisor through $x$ on $X$ is going to be linearly equivalent to a divisor not passing through $X$. In particular, it's clear that there exists a set of generating divisors of the type you want.
On the other hand, if you have divisors on $\hat{X}$ that satisfies those two conditions, then the images (pushforwards) of those divisors also generate the divisor class group of $X$. In particular, the divisor class group of $X$ is generated by divisors not passing through $x$. In particular, it is generated by Cartier divisors (since $x \in X$ was the only singular point). This implies that $X$ is factorial.