The tool is called an "edge slicker", and it is a polished concave groove, usually around the perimeter of a round plastic or nylon disc. Though a highly polished rod will give you the same effect, with slightly more work to make it look nice. I like metal (being a metalhead), but plastic or glass or anything smooth will work. That leather looks to be chrome tanned (just guessing from the appearance) and a wax might work better for slicking the edges than water.

That's a very nice first effort. Good job !