I am trying to grok this subject. I've gone through some old posts on the FoF forum to try to work it out for myself. Here is what I think happens:

Regular unit stationary in solid cover: gets in-cover and solid cover bonus = +2 defence dice
Regular unit stationary within 2 inches of solid cover: gets in-cover bonus = +1 defence die
Regular unit moving in solid cover: ??
Regular unit moving within 2 inches of solid cover: ??
Regular unit stationary more than 2 inches from any solid cover: gets in-cover bonus only on passing quality check otherwise "exposed" = +1 defence die if passes QC
Regular unit moving more than 2 inches from any solid cover: gets in-cover bonus only on passing quality check otherwise "exposed" = +1 defence die if passes QC

Irregular unit stationary in solid cover: gets in-cover bonus only on passing quality check and solid cover bonus always = +1 or +2 defence dice depending on QC result
Irregular unit stationary within 2 inches of solid cover: gets in-cover bonus only on passing quality check = +1 defence die if passes QC
Irregular unit moving in solid cover: gets solid cover bonus only = +1 defensive die
Irregular unit moving within 2 inches of solid cover: gets nothing but is not "exposed"
Irregular unit stationary more than 2 inches from any solid cover: is "exposed"
Irregular unit moving more than 2 inches from any solid cover: is "exposed"

Am I the only one bewildered by this stuff?

You've basically got it, outside of a few misconceptions. Let me try to clear this up:

If you are moving inside of or stationary in solid cover, you get a Solid Cover bonus. Think of it like moving down a stone wall or standing at one; it'll cover you as long as you are against it.

For In Cover, you can only attempt it if you are standing still. When I game with unfamiliar players, I call it "Taking Cover" or "Going Prone". Essentially it's your guys making best use of the cover around them, and they can't be moving to do that.

For Being Exposed, it doesn't need to be solid cover or not. It could be bushes, scrub, any terrain piece which could provide concealment; it might not block a bullet, but you can make yourself harder to hit. So while bushes might not give you a Solid Cover Bonus, they prevent you from being exposed to the enemy.

So to give you a full cheat sheet list:

Regular Forces
In Solid Cover
Stationary: +1D for Solid Cover, +1D for In Cover Bonus
Moving: +1D for Solid Cover

Within 2" of Cover or Concealment
Stationary: +1D for In Cover Bonus.
Moving: No bonus, not exposed.

Outside of 2" of Cover or Concealment
Stationary: Exposed. May get In Cover Bonus if QC is passed.
Moving: Exposed.

Irregular Forces
In Solid Cover
Stationary: +1D for Solid Cover, QC to get In Cover Bonus.
Moving: +1D for Solid Cover.

Within 2" of Cover or Concealment
Stationary: QC to get In Cover Bonus.
Moving: Not Exposed.

Outside of 2" of Cover or Concealment
Stationary: Exposed.
Moving: Exposed.

Hopefully that helps!

Awesome, thanks.

So the in cover bonus and the exposed penalty essentially cancel each other out for regulars beyond 2" of cover/concealment?

Yeah: both sides are gaining a die, so theoretically they'll offset. It's an interesting tactical decision: Get up and go, risking getting caught in the open with the reaction test or stay there, hopefully take cover and weather the storm.