Rust doesn't contain tetanus itself. It comes from a bacteria in dirt and poop in the ground outside and its been well known for centuries. If you were outside and you cut yourself, I'd go get a tetanus vaccine at a local walk in clinic just to be safe but you could have a fever and negative reactions. Last time I got a tdap vaccine was 3 years ago and I'm extremely healthy.
That wound doesn't seem to be very deep, considering how much it isn't bleeding. Usually a deep cut is going to be absolutely gushing with blood. It's where you can see the yellow fat blobs in your finger, that needs stitches and even that cut will be fully healed in 14 days and I know because I had my finger cut open while opening a metal ring I was trying to pull apart in my kitchen that was razor sharp. Cut my finger wide open and I could see the fat, It didn't hit bone though but I had blood all over my hands shorts and when I got home I had to clean up my floor because it was covered in blood.
Tetanus has been around since the 1700's with confirmed reports and probably older. Here's a painting of a man suffering from tetanus spasms, painted in 1809.
https://en.wikipedia.org/wiki/Tetanus 