I accidentally cut a very small part of the first cover (the part where the logo is) of a battery with my nail while trying to get it open. Is it still safe to use?

Well, I think it is okay to use if it was only a little scrape, but if you want to be safe buy a new one. But a small scrape would not cause much damage. SO it depends on how pig the scrape on the battery/puncture is. But to be safe if you can buy another one to be safe.