The German language is called complicated not without reason! I will try to set some things straight. There are three very similar words, two of them spelled even the same, but with different meanings. All three are related in origin to the English word "stop".
1. stützen (with Umlaut) = to support, to prop up, "stop from falling down". This is the word behind Brenecke's "Stützhakenverschluss". In the context of gun barrels, Haken = hook is the technical term for the things called barrel lumps/lugs in English. So a "Stützhakenverschluss" may be translated as a "supporting lump action".
2. stutzen = to trim. dock, clip, shorten, "stop from growing". Lancaster is right, the Austrians called any rifle shorter than a musket a Stutzen. The Steyr factory even called the 24" barreled, half stock iterations of the Mannlicher-Schoenauer "Stutzen" pre-WW2. In most of Germany, except the southernmost part, this term was unknown until after 1900. Only after the Austrians started to sell their M1903 Mannlicher-Schoenauers, "Stutzen" became known. As the most popular version was the full-stock carbine, "Stutzen" became the German household word for a full stocked rifle, just as "Mannlicher stock" in English.
3. stutzen = to hesitate, to become suspicious, "to stop short". Rarely gun related, except in some hunting situations when something unexpected is noticed. May also happen if you realize that you packed the wrong cartridges F.I.