What does the Ricoh WG-M1 “Mode Settings” menu option do?

Wow, that is a confusing section of the manual. In fact, it's a confusing manual overall — I think partly because of poor translation to English. For example, it uses "viewfinder status" to mean "when the camera is in the state where the LCD screen is acting as a viewfinder", as opposed to "menu status" when it's showing the menu, or "playback status" for ... well, playback. That's an odd use of "status".

Likewise, "Mode Settings" isn't really settings. It's just one setting: which mode do you want to be in. It appears to simply be an alternate way of toggling between movie mode or still photo mode. The other way is explained in the note on the bottom right — when in "viewfinder status" (that is, not in the menu or playback), you can just press the OK button (the big round one) to switch back and forth.

Unfortunately, as you note, the menu does not change the default. Since this is an "action camera", it's biased towards video, and there's no way to make still mode be the default. (Reference).