Wait, so it's both?
Originally Posted by Saito Hikari
Interestingly enough, they decided to take a page from BG3 and change Cutting Words to work on all of a single enemy's rolls for that turn. [...]

The important part is that it is still triggered as a reaction, so the potential for multiple prompts per turn is still there. As a reaction though, this means that when you do use Cutting Words, it is for a relevant enemy roll, while the BG3 pre-cast version runs the risk of an enemy just not doing anything that Cutting Words affects.
So it's triggered as a reaction: anytime an enemy you can see within 60 feet makes a successful attack (that Cutting Words could prevent?) or a damage roll, you get a prompt to use Cutting Words? And then that enemy subtracts 1d6 from said attack & damage roll, and any other attack and damage rolls during that same turn?