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 $X$).

[UMich dissertation][1]

**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.

  [1]: http://deepblue.lib.umich.edu/bitstream/2027.42/60794/1/ofelguei_1.pdf