2152hq, none of the rifles you mention was designed or made in Steyr. "Steyr" in the first place is a city in Austria. Josef Werndl founded his gun factory there in 1864. The company was converted to a joint stock company named "Oesterreichische Waffenfabriks Gesellschaft AG, Steyr" = Austrian Arms Manufacturing Company plc in Steyr, conveniently shortened to OeWG Steyr or "Steyr" only. F.von Mannlicher became chief designer, later director of that company. About 1900 the OeWG Steyr was the largest gunmaker worldwide, supplying many armies with rifles and carbines of many different designs and models. So, If your rifle is marked "Steyr 1904" on the left receiver wall, it is one of their other models made in 1904. Steyr inscribed the model year on the receiver ring, often scrubbed off by the British importers who sporterized many M92-M95- M1900 -1903 Mannlicher and Mannlicher-Schoenauer rifles from the military production runs.
The "Mauser Mod.1904" you found in the military rifle section of the 1911 ALFA catalog, page 17, is an entirely different action. It is the Mauser-Vergueiro, made by DWM, Berlin in large numbers for the Portuguese army, cal. 6.5x58P, and in small numbers for Brazilian military police units, cal. 7x57.
The Haenel sporters discussed here were made neither in Steyr nor in Berlin, but in Suhl, Germany. You are right, the 1911 Alfa catalog shows on pages 454 and 455 seven grades of sporters on M88 actions and on page 460 three grades of the Haenel "1909" rifles.According to an original 1909 Haenel catalog these are the lower Haenel grades Nr. 5,6 and 7 only. If you look close, the ALFA pics still show the 1900 model. It was expensive to produce new cuts for a catalog, not worthwhile to show "minor" differences.
The designation "New Model Mannlicher -Haenel" for the flush-magazine Haenels was an invention of the American importers, f.i. A.H.Funke, New York. The monicker "Neues Modell" is a re-translation, never used in Germany.

Funke and Tauscher already imported Haenel rifles on M88 actions for several years. As the M88 action used a Mannlicher-type 5 round enbloc clip, the importers called these rifles "Mannlicher Haenels". After Haenel brought out the flush-magazine M1900 they simply called this the "new model". Some US ads even invented a "model 1902" just to show that it was a new item they offered. My old friend Lud Olson knew only such American ads for the "New Model" and the "model 1909" designation from the ALFA catalog. As he had seen only three such rifles in all his life, he was not aware of the fact that there were two different Haenel models, 1900 and 1909. In his book "Mauser Bolt Rifles" he even wondered why the model name was "1909", but such rifles were advertized some years earlier. I found out about the real model history from German sources and told him the facts, but Lud passed away before he could rewrite the story for future editions. As most American gunwriters take Lud's book as gospel the designations "new model Mannlicher-Haenel" and "Model 1909" stick in American literature.