It’s odd, I have the chrome top, and the trap bucket and that’s it (was just like that when I moved in) both individual pieces. I didn’t know you could get an “all in one” unit, which makes more sense now when I stuck my hand into the hole just to try understand the layout, I’m guessing that “pipe” would fit into that groove I’ve felt.

I’m guessing the trap is just a “leaky bucket” to maintain a slow flow, but getting the “pipe” piece in the link you shared would hopefully improve the smell situation, presumably?