I'm perplexed: can someone explain to me how different derailleurs in my specific situation would affect shifting ratios?

When I bought the Jtek Shiftmate #3 last year I had 9-speed (post-2001) Campy shifters running to a Shimano Sora derailleur and a 9-speed Shimano Ultegra cassette. I did the right thing, according to these two authoritative sources:

https://jtekengineering.com/shiftmate.htm
https://www.sheldonbrown.com/harris/shiftmate.html

However, the shifting never worked very well at all, and since swapping out the Sora derailleur for a Centaur derailleur nothing has changed--it's inaccurate and very unsatisfying. (Yes, I will eventually buy a Campy conversion hub body and cassette for my Dura Ace hub--it's an evolutionary process.)

Nevertheless, according to the chart my current Campy-Campy-Shimano setup would require a Shiftmate #1, and an LBS tech guy has confirmed that derailleurs do indeed matter in this situation.

Question 1: How exactly do my shifters know or care about whether I have a Shimano or a Campy derailleur on there? Aren't derailleurs basically "slaves"? I don't get it.

Question 2. And furthermore, if the answers to question 1 are "They don't" and "Yes," then why would there be two different Shiftmates for the Campy-Shimano-Shimano and Campy-Campy-Shimano scenarios?

If you read the instructions, you'll see that the cable comes in on one size pulley, crosses over, and leaves on a different size. This changes the in and out cable travel. Each index shifter needs to match the rear derailleur and this thing translates the ratio between different models. If Shiftmate doesn't have a model for your new setup, you can always go back to basics, friction shifting!

Your shifters don't know what kind of derailleur you're using, or vice versa. The only thing your derailleur knows is how much cable your shifters are pulling.

Your Campy 9sp shifters pull 3.0mm per click, and your Campy derailleur will move 4.55mm (Campy 9sp spacing) for each 3mm of cable. However, your Shimano 9sp cassette has the cogs spaced 4.36mm apart.

Campy and Shimano have different mechanical advantages which is why they aren't compatible, except some special cases. The shift-mate is designed to match the three variables: cable-pull, RD mechanical advantage, and cassette spacing.

Here is a great website that has a lot of data about mixing and matching: https://www.ctc.org.uk/DesktopDefault.aspx?TabID=3946

I never really got around to messing with any of these combos, I just decided to run friction shifters and be done with it. So I have Suntour shifters, Campy Veloce RD, Suntour FD, Shimano 7sp cassette, and SRAM 9sp chain!

Derailleurs are "interchangable" only within a given manufacturer's product line. I.e. you can use any model Shimano 8-speed (except Dura Ace), 9 or 10-speed rear derailleur with Shimano shifters. Same for Campy within in a given period.

However each manufacturer's derailleur line has specific geometry in the parallelogram design and do not move the same amount as other manufacture's derailleurs for a given cable pull.

The Shiftmate corrects for those differences. You do have to have the correct model and install the double pulley in the correct orientation.

Ah! That the derailleur does not move the same distance as the cable moves, I did not know. I had thought that any derailleur simply moved as much as it was pulled (and I swear I've seen some tech talk to that effect somewhere before).

I guess I need the Shiftmate #1. Thanks very much for the helpful replies!