OK, so just to make sure we're all on the same page. Left and right are from when looking to the front from the rear of the bike. Is that how you're referencing L and R? And to just make it extra clear, the photo in post #7 where the shifter says "9 Speed" on the top of the lever, that is considered the Right Shifter.

Generally countries that drive on the left, and Italy, have front brake on the right by default; some riders like to switch but that's how new bikes and hire bikes are supplied, unless specifically requested otherwise.

That is correct.

Disconnect the cable from the derailleur and see how many clicks you have. The shifter that says "9 speed" should have at least 8 clicks

I think you're entirely correct. Thx.

Done. 3 clicks. Up and down.

sounds like your RHS shifter is malfunctioning and needs an overhaul. Personally I’d send it off to someone, although parts are likely available if you want to do it yourself. A 9sp LHS shifter is ratcheted, rather than strictly indexed, so it might well have 10 clicks. That’s how my 9sp Chorus LHS operates, although I’ve never counted the clicks. It takes 3 clicks of either the big lever or the thumb lever to shift chainrings, and the limit screws keep the FD in line- that’s all I care about

Thanks a ton for everyone's advice.
Since I had eventually set up the rest of the drive train as 10-speed, I solved my immediate issue by simply putting on Campy Record 10-speed shifters. These are working beautifully and I can now ride this beauty.
I'll tear down and rebuild this 9-speed set.

I'm very used to building up older group sets Suntour, Gran Sport, Nuovo and Super Record, etc. and have never dealt with Campy's brifters before. Shimano shifters have never given me any problem
I've learned a ton with this pair, most of that the hard way, which is sometimes the best way.

All done, all 10-speed Record, shifts like a dream.