Usage of the phrase 'make one's eyes bleed'

Both your example sentences are perfectly fine and exactly how people use the expression.

The Urban Dictionary definition is badly worded (if you’re just getting started with using Urban Dictionary, get used to this—most entries on there are horribly worded). What it means is that you can also say something like,

“Do matrix operations? Gah, I’d rather bleed from the eyes!”

This emphasises that even bleeding from the eyes is more appealing than doing matrix operations.

I wonder where the Urban Dictionary gets some of these definitions... I think they should hang out with some people who are over 25 once in a while.

Logically, "bleed from the eyes" is about seeing something unpleasant:

"My grandmother showed up in a bikini and I started to bleed from the eyes."

The most common phrase (US) for not wanting to do something is still (the tried and true) "I'd rather have (a) root canal."

"Go to prom with Mildred Goggledorf? I'd rather have root canal."

The only time a task would cause bleeding from the eyes, is if was long and/or arduous, and involved strain on the eyes.

"I tried to proofread that idiot's report, but I had to stop because my eyes were bleeding."

"The Staff Sargeant made us do push-ups until our eyes were bleeding."

The only other "bleeding eye" usage I'm familiar with is when you want to inform someone that it is perfectly obvious that they were smoking marijuana before the big meeting - but you don't necessarily want to inform everyone else there. In which case you would stare into the person's face and say, "Close your eyes before you bleed to death." (A reference to their characteristically bloodshot eyes.)