Tetanus spores are pretty much everywhere, and can survive dormant for literal years. But it can’t survive in aerobic conditions after blooming, which happens when it comes into contact with blood. So surface cuts aren’t really an issue regardless of what caused it, because the bloomed spores will quickly die in contact with air.

The association with rust is because rust is historically very good at harboring tetanus spores; This is regardless of whether or not an animal has shit nearby recently. It has a lot of jagged surface area for the spores to hide in, and punctures from rust tend to be accidental and deep enough to force the spores into the anaerobic tissue. Again, this can happen in seemingly shit-free dirt, because the spores can survive for so long that they’re basically present everywhere that has dirt.