You're not clarifying anything, rather the opposite.
There are not two different 'greedy' here, greedy has only one meaning:
A greedy regex matches as much data as it possibly can, while an ungreedy regex matches as little data as it possibly can.
that's all.
Tomao: /g has nothing to do with greedy and /U only makes the whole expression ungreedy by default, where .+? would now represent a greedy search