The AISI/SAE standards and nomenclature were not developed until the 30s and 40s; long after these barrel steels were manufactured. This is a helpful summary
https://www.azom.com/article.aspx?ArticleID=6151
The standards provide a range within which the AISI number may be assigned.
AISI 1040 can have carbon .37-.44% and manganese .60-.90%
That allows for variations in steel batches.
The metallurgists at METL labeled MOST of the samples I submitted as "non-standard" ie. one chemical was out-of-range or the composition just did not fit. METL’s analysis of a 1898 Hunter Arms Armor sample: “The measured results are comparable to 1211 rephosphorized and resulfurized low alloy steel (UNS G12110) as well as 1045 plain carbon steel (UNS G10450).”
Said another way is 1045 with too much phosphorus and sulfur, which make machining easier but lowers ductility and impact resistance; and other stuff
https://www.linkedin.com/pulse/21-chemical-elements-effects-steel-mechanical-properties-jeremy-he

A 1908 Hunter Arms Armor is non-standard AISI 1018 with slightly high phosphorus and sulphur, and a low concentration of nickel.

A c. 1925 Crescent Fire Arms “Genuine Armory Steel” non-standard (high phosphorus) Alloy AISI 1040 Carbon Steel with a low concentration of nickel.

Bottom line: We can KNOW what some barrel steels were because the makers said so, or because samples have been analyzed. It is not unreasonable to then assign AISI numbers for comparison, and because this happens to be of interest. And AISI 1020 steel made in 1900 is AISI steel made in 2020.

I think Dave Suponski's analysis of Parker steels is most interesting.
Titanic, post-WWI Vulcan, and Trojan were essentially the same; AISI 1030 and 1035 Medium Carbon steels. Titanic did have low levels of both nickel and chromium compared to the others, but it would not be an “alloy steel”. "Parker Steel" turned out (surprise) to be Decarbonized Steel, as was "Remington Steel", according to Remington.
More on Bessemer process/Decarbonized Steel to follow.

SO (again) other opinions as to what Winchester Nickel Steel and Stainless Steel were are most welcome.
I think Mike Hunters statement regarding the composition of Nickel Steel (probably 2330) and Proof Steel (4140) are definitive. I've not yet heard back from him.

And I at least have learned something from this exercise.
I'd also like to understand why "the near certainty that 304 stainless was not used" is nearly certain? What were the other stainless steel options c. 1926?
Krupp's austentic stainless developed about 1912 with .04% carbon, 16.5-18% Cr, 10-13% Ni?
Krupp “Nirosta” 1912 patent NIchtROstender STAhl 21% Chromium / 7% Nickel stainless steel introduced in 1913?
Bohler “Antinit” Rostfrei Laufstahl chrome-molybdenum-vanadium introduced 1912?
Haynes' 1919 patent martensitic 420?