I'm not 100% sure but your definition might be equivalent to AoC. Though, the official definition is without the disjoint set condition. The point of AoC is that it allows you to choose one element inside a set for each set in some collection of sets. The only condition the collection has to satisfy is that it doesn't contain empty sets.

Basically, we just want to choose things (via a function) when we know we can (when the set we're choosing from is non-empty).

I'm not sure what you mean by "construct a new set". If you mean that the image (range) of a choice function should always be a set that's not in the domain, that's interesting, but I've never heard of it. Though, there is one trivial case when this couldn't work - collection of sets being e.g. {{1}}, which gives us only one choice function: {1} -> 1, so its range is {1} which is inside {{1}}.