Sharptail said:
... I am still puzzled as to why this gun is labeled as a Mk. III. ...
The confusion you are experiencing arises from the fact that the "Mark III" designation refers to the version of Martini-Enfield Artillery Carbine this Martini-Henry was converted into ...
not to the pattern of Martini-Henry Rifle it started out as.
As I mentioned, the A.C. Mark III differed in subtle ways from Mark I or II versions - one evident difference would have been that it would not have had any provision for the clearing (cleaning) rod - i.e. no hole through the nosecap - as the conversion process which produced your carbine occurred in 1899, after clearing rods had been abolished on all .303 rifles and carbines. (Because your carbine has been altered, that difference wouldn't be apparent, of course.)
One thing which could maybe help in understanding this is that all of the markings relating to the original production of this arm as a Martini-Henry should be on the
right side of the receiver body. For example, though you haven't shown the original markings on that side of your reciever, you do mention that your carbine was originally produced as a Mark I Martini-Henry, subsequently upgraded to a Mark II. Assuming that the right-side markings are true to form, that fact would be evident from the original "Mark" designation having been a roman numeral one, usually followed by a period - i.e. "I." - which would be
centered directly below the other markings. When upgraded to Mark II specifications (as happened to almost all Mark I Martini-Henry arms) the second "I" of the roman numeral was added just to the right of the first "I" (usually partially covering - though not obliterating - the period) so the new Mark designation then appeared somewhat like "I.I", slightly off-center to the right. Is that how your markings appear?
The next part of the puzzle to keep in mind is that any markings regarding conversion of a Martini-Henry to .303 caliber (either Martini-Metford
or Martini-Enfield) were to be placed on the
left side of the receiver (i.e. the markings you have shown)
and that
all of these markings relate to the conversion and new designation as a .303 rifle or carbine.
One notable exception to these general rules are the "Citadel" Martini-Enfield rifles, which were converted at the Citadel Arsenal in Cairo, Egypt, using rifles and parts supplied by the United Kingdom. The
new markings do appear of the left side of the receiver,
but the original right-side markings were "scrubbed" - i.e. obliterated. That is another story ... the "short version" being that from about 1881-82 until the outbreak of WWI Egypt was almost completely within the British sphere of influence (e.g. British forces stationed there, a Royal Navy station maintained at Alexandria, and the Egyptian Army being trained, and largely officered, by the British, etc.) in order to safeguard Britain's interest in the Suez Canal. However, Egypt remained - technically, at least - a province of the Ottoman Empire. Thus it just "wouldn't do" for the Egyptian armed forces to have British-marked rifles ....

However, when Turkey entered the war on the "wrong side", all such pretence was swept aside and Egypt was declared a British Protectorate - indeed, the British Near East Headquarters was in Cairo throughout WWI ... remember "Lawrence of Arabia"?)