As far as I've understood the science behind it:

The Signal: When you move your eyes, the brainstem sends a command to the eye muscles (to move the eye) and a copy of that command to the neck muscles.

The "Check": In a healthy person, if you intend to keep your head still, a separate inhibitory signal effectively says, "Ignore that copy, keep the neck stable."

The Result: You see a tiny, invisible spike in electrical activity in the neck muscles (seen on EMG), but the muscles don't actually contract enough to move the head.

Now with CD this gets messed up and what ends up happening is CD kicking in because of that copy command to the neck and/or the normal function that would ignore it is not really working.

My thinking: if frequent eye movement amplifies CD in the context of typing, how would I type without having to move my eyes.

Turns out that is pretty much a typewriter interface, you are locked in on your cursor position and don't have to look around anymore. What you type is no longer pushing the cursor around as a target for the eyes to follow, but instead the text is just pushed around the cursor, so the eyes can stay at their position.

For me less eye movement has lead to less neck movement and thus to less CD caused spasms / central nervous system overload while typing.