which seem like sensible English descriptions if for some bizarre reason you can't use the shorter names

The motivation of Kahnmann to use system I and system II isn't to have shorter names. It's that there are existing conceptions among people about words describing mental concepts and he doesn't want to use them.

Wikipedia list from Kahnmann:

In the book's first section, Kahneman describes two different ways the brain forms thoughts:

System 1: Fast, automatic, frequent, emotional, stereotypic, subconscious
System 2: Slow, effortful, infrequent, logical, calculating, conscious

Emotional/logical is a different distinction then intentional/autopilot. Trained people can shut on and off emotions via their intentions and the process has little to do with being logical or calculating.

But even given them new names that scientists don't give them might be a valid move. If you how even do that then you should be open about the fact that you invented new names. Given science public nature I also think that you should be open about why you choose certain terms and choosing new terms should come with an explanation of why you prefer them over alternatives.

The reason shouldn't be that your organisation is named "intentional insights" and that's why you call it the "intentional system". Again that pattern leads to the rationality is about using system II instead of system I position with differ from the CFAR position.

In Gleb's own summary of Thinking Fast and slow he writes:

System 1 runs automatically and System 2 is normally in a comfortable low-effort mode, in which only a fraction of its capacity is engaged. System 1 continuously generates suggestions for System 2: impressions, intuitions, intentions, and feelings.

Given that in Kahnmann's framework intentions are generated by system I, calling system II the "intentional system" produces problems.

What is your credence for the explanation you give,

Explanations don't have credence, predictions do. If you specify a prediction I can give you my credence for it.