My two different controllers don't have the same button recognition ingame

The x360ce.exe does not work with all games. I recommend you look for the most updated drivers you find from the controls. It has happened to me with newer drivers that take a configuration very similar to the xbox controls. You could try, maybe this solves your problem. You also have the option of looking for another emulator of controls of xbox360. The x360ce is good but it is not the only one, try with another.

x360ce is probably not going to work for games launched through Steam if you just drop it into Steam's installation directory -- it's supposed to be in the directory of the game itself so that it's xinput*.dll can be loaded and then it can use the configuration you have setup. This is because Windows can/will automatically load .dll's that are in the same folder as an executable, considering them to be included with it. Part of SafeDLLSearchMode apparently (knew this happened, but never knew what the name was, lol).

Best idea would be to setup x360ce for each game...which obviously would not be ideal, but is likely what's going to be needed.

You could try messing around in Big Picture -> Settings (gear icon) -> Controller Settings turning some of the support on/off, but it's really not likely that this will allow the program to work just by being in Steam's own directory.

There are a few alternatives to x360ce, the most notable of which to me is Durazno, which is an open-source XInput to XInput (XBox 360 controller) wrapper that lets you customize them; sounds like a slightly-weaker x360ce from that description, but it may do the trick for you.