I always wear work gloves when running saws or other power equipment. Has saved or reduced damage many times. When you feel that little tug on your glove, you know it is time to remove your hand!
For example, when splitting wood one time with the hydraulic splitter, the splitter closed a split in the wood. When I felt it, I yanked my hand out just in time ... the finger of the glove ripped off and was crushed, but thankfully my hand was fine! If the glove had not given me some warning, I'm pretty sure I would have had some damage and possibly lost a finger.
Also, once with the chainsaws (a clutch spring broke and the chain spun w/o any throttle) and once with the hedge clipper (I extended it out with one hand and caught it with the other, my finger was off the trigger, but the blades had not stopped yet). Both times I had minor damage to a finger (one required stiches), but I believe in both situations the damage would have been much worse if I were not wearing the gloves. That fraction of a second it is going through the glove gives you some reaction time, and w/o a glove the damage to your hand would be that much deeper.
Also, I was glad I was at least wearing double layer work pants when using the chainsaw (just bought some more of them). I was tired, near the end of the day, and I reached out to cut a limb. My finger was off the throttle, but the chain had not yet stopped as I lowered the saw. It touched my thigh, the chain went through both layers of pants, but then "bound up". The pants looked horrible, but my leg only had a 4" scratch ... I lucked out!
Be careful out there, it can always be worse!