I've cut myself on an end mill that isn't even turning so I can imagine how something like this can happen in a flash. A vivid reminder to make sure any machine is powered down and stopped before making any adjustments or getting your hand near it. About 25 years ago I was removing and emptying the grass catcher on a gas powered mower with it running... My daughter came out with a trash bag and I was momentarily distracted and my hand entered the grass chute opening. I'm lucky and still have the hand and my fingers, but it looked a lot like yours (same hand, same finger). It's number one on my list of stupid things I've done in my life, but ever since then I'm extremely careful around any machine (not just lawn mowers lol). I imagine you will be too! A tough way for you & me to learn a lesson, but could have been much worse.
Hope you heal fast!