Still on the subject of carry on bags.....

The measure of the bag is including all "protuberances", which includes wheels and handles. Always. Every bag. On every airline. It will say so in the allowable bag information provided on the airlines' web pages.

Will the bag that actually measures 23" have to be checked? That depends on what mood the gate agents are in when you are boarding your flight. If the flight is booked full and everyone is dragging overstuffed carry on bags with them, expect to have to check yours unless you are one of the first on the plane and have a high level frequent flyer status with that airline.

The rules are the rules. And the rules state that wheels, handles, and whatever else your bag has, is part of what gets measured.

Whether or not they are strict with the rules is another question. Sometimes they are, sometimes they are not. Some people get away with skirting the rules (sometimes by a lot), sometimes you can't get away with anything. There's no real way to know.

Is it worth risking it? Depends on circumstances. What would the consequences be if they catch you and enforce the rules to the letter? On some airlines (typically the "low cost" carriers) they are very strict and have surprisingly high penalties for breaking the rules (that's how they make money). Some airlines are more relaxed. A lot may depend on the mood of the check in staff, and on how full the flight is (and how blatantly those ahead of you in line have violated the rules).

The only way to be sure you have no trouble is to follow the rules. Or, you can just try your luck and see how it goes.
Do you feel lucky?

Agreeing with Mark and David. What the luggage company called your bag is not relevant; it's what it actually measures when handles, wheels, and any other "protuberances" (great word) are included.

In Europe, they use sizer boxes. If your bag can be made to fit into the sizer, it can be carried on; if not, it has to be checked. So pretend you have to put your bag in a sizer of the dimensions the airline states. Squishing is fine, but fixed protuberances like wheels, handles and bulges are not. If your bag will fit into this, it's good. If not, as said above, whether or not it can be be carried on depends on the agent on duty.

And yes, when I went to buy luggage, I made sure to measure it with a tape measure, and my bag was called 20 inches, but actually measured 21 inches (so it was fine for my airline). It was a two wheeled bag; you are right that on spinners, the four wheels protrude more from the bag than the two wheels do on non-spinners.

A worse problem may be that 21" is the maximum that will fit in the overhead compartment. If the 23" bag does not fit what are you going to do??

If the airline has a 21" max., and most people will comply, a bag standing 2" taller is likely to stick out like a sore thumb, especially the 4 wheel spinner bags.

Every airline site I looked at states the wheels and handles 'count'.

My 'guess' is, it is more likely than not that a bag at 23" will get refused boarding

If they throw it in the sizing box -- it all counts. A non-wheel carry-on, even if overstuffed and often be stuffed into the box but a rigid, wheeled case will not. The difference between 21 and 23 is noticeable -- a half inch, maybe note. I would not chance it.

"My 'guess' is, it is more likely than not that a bag at 23" will get refused boarding"

Or possibly worse, you have to gate-check it when you get on board if it won't fit in the bin. If you decide to risk it, be sure all your valuables, important documents, electronics and chargers, and meds are in some kind of cube you can easily slide out if you have to gate check at the last moment. There is nothing worse than having to fumble thru an open bag trying to get stuff together to check a bag at the last minute.

How do I know? First trip to Europe. Flew Delta over, Delta back ...but didn't notice it was actually an Alitalia plane leaving Florence on the return and we had to go by their weight regulations for carry on. Size? Rick's 22" convertible backpack was fine. Weight? Uh, no. Yep, had to dig for my kindle, chargers, etc at the check-in desk. Whew, was I frazzled and embarrassed.

The truth is that not all airlines put all bags in the rack. It often depends on the mood of the airline personnel on that given day. Most of the time, a 23" bag will pass through airline eyes. We travel with 21" TravelPro bags.

Norwegian Air Shuttle in Oslo had half the passengers on the plane, and they decided to be difficult and put the remaining carry on's into their rack. All they did was take my bag and charge me $50 to put it in the belly of the airplane.

They could also get many passengers on 10 kg. max. When one gate agent pulled that on me, I offloaded some of my stuff into my wife's underweighted bag--holding up their line.

The 21" European size limit is just the airlines' standard. All of the airplanes will take a 23" bag easily on overhead compartments.

Your Air Dolomiti allowance is slightly shallower than the Air Canada dimensions and I can expect with their weight limit, nothing 23 inches is going to be accommodated. If this was a separate flight, I would just check that bag otherwise, I suggest a lightweight back pack....something with similar dimensions, but foldable meaning 'light'.
If you are going to buy new standard luggage, go to Bentleys in your mall, take a measuring tape and if you have one, a luggage scale...or borrow one of theirs. That is the only way to shop for what will meet your requirements.

Agteeing with Mark, David from Seattle and Harold. It's the out-to-out dimensions that matter. If your bag is too long (or too wide or too thick) it will not fit the sizing frame the airlnies use as a go/no-go guage. More importantly, if your bag is too long/tall (21" + 2" = 23"), the overhead compartment door will not close unless you reposition the bag lengthwise instead of wheels in. Your bag would then take up the space of two bags placed bottom in.

Read the CU article on carry-on size:
https://www.consumerreports.org/luggage/When-carry-on-luggage-isnt-carry-on-size/

Thank you all.
I did take my tape measure the day I posted the question, and went out and found what I hope is the perfect bag.
It's a Heys Spinner, and measures (with wheels) 21".
I did a test pack before I took the tags off, and could fit tons of stuff in it.
Plus , it only weighs 2kgs empty, which is even better!