You have hit all the identification marks 'right on', friend. It will be one of several models, all designed by the Ritter Ferdinand von Mannlicher and mostly built at Steyr..... but not all.
The series started with the Model 1885, a .43-calibre single-shot. This was modded to a repeater and produced as the Model 1886, also in .43. But this was right when smokeless powder was starting up, so they modded it again as the 1888, this time to a 8x50R using compressed Black powder. By 1890 they had a smokeless powder which would work, so they had to mod the rifles again because the new powder shot flatter than the Black, meaning that the sights had to be changed...... and it became the 1888/90.
ALL of these rifles had a prop-block straight-pull action which reminds one of a BAR turned upside-down mechanically.
So von Mannlicher came up with something better: a rotating-bolt straight-pull with controlled-round feeding like a Mauser and a new action, rather on the lightweight side, designed for the new powders and cartridges. This was the 1895 in the infantry rifle, and these are the ones you see most of the time. Some of the older models were sold here back in the early 1960s but have been scarcer'n turkey-teeth ever since.
The 1895 was the front-line rifle of the Austro-Hungarian military in the Great War but they also were purchased by Chile and as the Model of 1903 by the Kingdom of Bulgaria. Bulgarski rifles are marked as "STEYR 1903" on the left-hand receiver rail and with a Royal Bulgarian crest on the receiver ring: very pretty. Of course, by the time the Great War got started in 1914, these rifles already had been spread around by the First and Second Balkan Wars, so everybody had a few but Austria-Hungary had the most an they had the only two factories for making more.... which they did, at Steyr and at Budapest both.
Following the Great War, a lot of AH rifles were spread around to the successor states of the old Empire and more went as war booty, so they were used again by Austria, Hungary, Bulgaria, Czechoslovakia, Yugoslavia, Greece, Italy and possibly a couple more. And Chile still had their little supply, too.
In the 1930/31 period, most of the existing Austrian, Hungarian and Bulgarian rifles were modded to the 8x56R30M cartridge, using a .330" bullet of a mere 208 grains weight (supplanting the older .323"-244: now, THERE was a kicker!) and marked with a 12mm letter "S" on the chamber. Quite a lot were modded, mostly in Yugoslavia, to the 7.92x57JsS German cartridge. These were stamped "M95M" on the receiver ring.
So there is a LOT of variety in these rifles and they are ALL collectible and they are ALL fun to shoot. By determining exactly what you have, you can know their history.... or most of t.
But it sounds as if that is what you are dealing with.
Have fun!
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