Given the pattern on your razor, I really don't think it's the leather that caused the problem. Long term, leather can cause rust. But that quickly is from not properly cleaning the blade. Even if a razor looks clean, there can still be soap residue from the lather, and that looks like you wiped the blade but didn't quite get it all off. And, BTW, if it is from soap residue, putting something else on the blade (Tuffglide or oil) won't help UNLESS you remove the residue in the application process.

Personally, I rinse under the hottest water I can get from the tap, then wipe the blade off with toilet paper a few times, each time making sure the TP is dry or getting a new square. I hold the TP in my hand, lay the razor spine down, pinch the razor, and draw from heal to toe. Pinch tight, and be sure you get the whole blade (including the edge).