So I bought a headset along with a frameset, and didn't need to check for suitability as the seller did that for me.

Alas, a LBS cut my steerer tube too short while attempting to install the head set, and now I need a new headset with a lower stack height.

It's a one-inch threaded steerer, 24 TPI.
If it's relevant, the frame & forks are Columbus steel Fausto Coppi of around '95 vintage.
The outer diameter of the head tube is 31.8 mm, the inner I can't measure since there is a headset in place.

Now, stack height in this case is the total height of all the headset components that are left visible outside the head tube, right?
Measured like this, my current headset has a stack height of 40.00 mm, without any washers/spacers in place.
As it sits, the locknut doesn't go deep enough, the steerer tube is about 2mm too short even without a washer in the headset.

I know I will be needing a one-inch threaded headset - one with 38 mm stack height, including the locking washer- but what about
crown race diameter, S.H.I.S measurements all other measurements ( I don't even know what S.H.I.S stands for) ?

So what measurements am I going to need in order to choose the right headset?

I would love a Campagnolo headset, and one could be had reasonably on the web, but I'm afraid the 36.5 mm stack height might be too much. Does the Campy stack height measurement include the washer below the locknut? I already have a beautiful Campagnolo engraved toothed washer I'd like to use, but it's 4 mm thick, which would likely mean the lock nut would not have enough thread to lock into..

And the LBS didn't offer to replace the fork that they cut incorrectly, or find a suitable headset at the very least???

Why are you searching for a new headset? If your LBS made a mistake in cutting the fork, it sounds like they should be the ones to solve the problem. And I know I would be expecting them to obtain a replacement fork, not a new headset.

I do not trust them to do the job right the next time, and these forks are no longer available - I took the compensation in cash and now I need some help in choosing the right headset..

I'm confused: you say you need a 38mm stack headset but the Campy at 36.5mm won't work? Why not? Is it that 4mm thick spacer that's the problem? If so, get rid of it and use a thinner one; the one you have is likely from a Super Record headset and wouldn't be appropriate with a different model anyway.

Well, the reason for asking is there within my original question; I don't know whether Campagnolo's stack height measurement of 36.5 includes a lock washer or not. Maybe one of you people can tell me? If it does not, then finding a suitable washer may be difficult as that mentioned 2 mm of height room isn't exactly accurate; it may be a couple of tenths either way. And then, if it happens I have to cut the steerer again by just a couple of tenths to make that new Campy headset fit, things can get ugly for the steerer. I'd rather have a headset that would be certain to have enough room for adjustment by spacers.

Anyway, what is the significance of crown race diameter and that elusive S.H.I.S ?

How do I know which headset will fit? Are all 1" threaded ones generally suitable or do I need one with specific crown race and other dimensions?

What model headset is this? If you don't know, a picture could help identify it.

Your ~1995 Coppi is almost certainly designed for ISO dimension headset parts. The only variable would be stack height.

The headset I was looking at, to replace the Miche that is now too tall, is a 2013 Campagnolo Record 1" Threaded Headset. From the pictures I have seen, a washer is not visible, but I guess that doesn't necesserily mean it isn't there.

About ISO sizing, what would that mean in millimeters? The usual information I see is crown race diameter, and obviously all the parts must fit to the head tube's inner diameter... But I guess that would be a standard diameter on '90s steel frames? Mine is 31.8 at the outside, but, I can't get the inner diameter just now.

The Campy height does not include any washers, so 36.5 is the minimum height..

ISO crown races have an ID of 26.4mm, OD of the cup flanges is 30.2mm

I'm confused. Your steerer is 2mm too short to work, or 2mm shorter than optimal? Because 2mm short of the locknut lip should be no big deal.

You need to know the dimensions of the headset as well as the stack height. This is useful: https://sheldonbrown.com/cribsheet-headsets.html

Good, thank you guys.

I'm still confused about what I need to do. Now there is no washer or lockring between the upper race and the locknut, and even tough the headset is set too tight at the moment,
there still is at least 2 mm of unused thread on the locknut.

So I'm really not sure if this setup can ever be safe, or work as intented.
Also it's an aluminum headset, so the locknut is alloy which means it requires more threads for a safe fit, than a steel one would.

I still don't know if I should try to make the current headset work, or just get a new one with lower stack height.
I also don't like how a washer cannot be fitted now, as the headset is too tall for that.

Maybe I am wrong, and others can correct me if I am, but the lock nut just keeps the top race from unscrewing. It is not a heavily loaded part, the steer tube takes all the stem load and transfers that load to the bearings/races which transfer the load to the head tube.