Do NOT put down the limestone. Take the soil test for what it is. Moss can grow anywhere there is dampness and shade.

pH is probably the easiest to measure accurately but home test kits are not that fabulous. You would do better to get a real lab test. Also, check with local nurseries, landscapers, gardeners to find out what the soil pH normally is around your area.

what if we have moss in areas that are not damp or shady, that struggle to grown grass?

It is probably poor, thin soil. It needs some sort of organic matter. Alfalfa, corn meal - whatever feed is cheap locally. The lawn gurus will probably be around with more specifics.

I have a pH of 7.2, tested by Cornell more than once. I also have moss in spots in my front lawn, where it gets almost full sun. The soil just isn't very deep or fertile.

Soil test kits are notoriously awful. I've seen claims that they can be used correctly but I remain dubious due to personal experience with the allegedly trained.

Send soil off to be properly tested through Logan Labs. For twenty bucks (basic test), you'll find out every major resource your soil has.

Even if, theoretically, your soil test kit returned a picture-perfect pH, a pH tells you very little indeed. It tells you nothing about what resources the soil requires to rebalance it, and nothing in terms of how much calcium, magnesium and potassium need to be added to correct the problem because it also tells you nothing about your soil's capacity to hold resources.

The balance of all three impact your soil pH, and all three need to be in the correct proportion for your grass (or any other plant, for that matter) to flourish.

Honestly, if there were one thing I could get rid of, it would be the concept that pH were important, or had any real meaning at all. It really doesn't, and when reading tests, I tend to mention it because people expect me to, but otherwise defer and say that I'll be discussing it under EC (exchange capacity), Calcium, Magnesium, Potassium, and Sodium (the latter does impact pH, but is rarely significant outside desert regions and, if it is significant, should be abolished).

Seriously, the discussion of pH tests should be started with the simple commandment, "Thou Shalt Not because it tells thee nothing good."

Don't get me started on types of limestone. Just don't.

Mosses tend to flourish in poor soils that have sufficient moisture (acidic, alkaline, they really don't care that much and I get mosses growing right on native limestone rock, pH about 7.8). They're weak, can be displaced by almost anything else, but until the initial poor soil issue is corrected they're going to dominate. It was an issue I had when my soil was in full collapse (nearly no resources) and that disappeared once I repaired it.

Barring repairing the initial problem, there's no sense killing or doing much of anything about the moss.

I am in a location where I cannot get a proper professional soil test. We have moss in many areas. Some are around trees (forested areas), some are in lawn areas (actually a different property). All are moist, the lawn areas also have some shade although a lot of sun (in season), the treed areas very shaded.

I'm not so concerned about the forested areas, but I have the following question - what are the usual most critical nutrient deficiencies related to moss? Are they general 'all the soil is poor' or would be best to focus on macro or micro nutrients? To be clear, I don't mind much the moss in the treed areas, although I am adding as much as I can of organic materials because the trees clearly absorb/consume as much as they can, but whatever. For the lawn area though, I would consider adding some proper fertilizer and don't have the organic materials in sufficient quantity to add nor will just adding compost/mulch a bit at a time likely work.

Or put differently if I found out adding miconutrients (iron / magnesium etc), I could do that easily, or a standard NPK mix in due course. Note, I'm not trying to eradicate moss here either, just improve conditions for the lawn so that lawn outcompetes in the medium to long term.

I am in a location where I cannot get a proper professional soil test.

There is no such location. Logan Labs will accept soil from the US and Canada. If you're in Mexico or Europe, other agencies exist that can handle it. Great Britain, same.

what are the usual most critical nutrient deficiencies related to moss? Are they general 'all the soil is poor' or would be best to focus on macro or micro nutrients?

There is no answer to this question. A soil can collapse on any major nutrient (and a few more minor ones), or for reasons of very poor general care. Or mosses can simply be a symptom of a lot of shade and a damp soil with rock under it.

You can undoubtedly find plenty of people who will give you any number of answers to this question. If they lack the chemical and biological knowledge to answer but are repeating what they've been told, that's forgivable. If they claim education and knowledge but answer anyway, that's incredibly unethical.

Mindlessly tossing a resource at a soil that may or may not require it doesn't make any sense, is as likely as not to waste money, and may very well be leaching into the water table where everybody else gets to deal with it. For some things, that's not too bad. For others, that's downright irresponsible.