Most people don't realize how messy blood is. I accidentally cut myself one Christmas opening presents. I got a new knife and was using it to open some zip-ties and it ended up stabbing my thumb on the inside under the knuckle and almost popped out the other side from under my nail. It was pretty deep and i didn't want to bleed all over everything so i ran to the kitchen without telling anyone.
I was in panic mode more because i didn't want to mess up my Grandmas house than because it was possibly serious injury. hahaha The good ol days. Thankfully my sister, who was in nursing school at the time, came and did a cool trick to help me from bleeding out!
She wet down a paper towel and had me apply it with a lot of pressure. The moisture in the paper towel (i guess you could use a rag or cloth too) made it to where the blood didn't have anywhere to flow. I don't know if it would work completely if you are on the blood thinners, if there's a next time, but it couldn't hurt.
Something about drawing in blood felt wrong. Even after it was finished. And some of the looks i got when i told them that it was in blood ink were priceless!