In German there is the name "Buntmetalle" which means "colorful metals" for non-ferrous metals which are also not noble metals: Cadmium (Cd), Cobalt (Co), Copper (Cu), Nickel (Ni), Lead (Pb), Tin (Sn) and Zinc (Zn) I didn't find any equivalent for this in English. Is there such a labeling/categorizing in English at all?
I don't think it's ideal, but there is the term "base metal". There are various definitions for what is considered a base metal, but the main noble (or precious) metals are always excluded. Some definitions also exclude ferrous metals. Unfortunately, this variation in meaning means it might be better to just say "non-ferrous, non-noble", as TheSimpliFire suggested in a comment. If you want to use "base metal" as a term for this specific set of metals, it's probably best to define it yourself, or at least refer to a definition.
Wikipedia states (emphasis mine):
In contrast to noble metals, base metals may be distinguished by oxidizing or corroding relatively easily and reacting variably with diluted hydrochloric acid (HCl) to form hydrogen. Examples include iron, nickel, lead and zinc. Copper is also considered a base metal because it oxidizes relatively easily, although it does not react with HCl.
In mining and economics, the term base metals refers to industrial non-ferrous metals excluding precious metals. These include copper, lead, nickel and zinc.
The U.S. Customs and Border Protection agency is more inclusive in its definition of commercial base metals. Its list includes—in addition to copper, lead, nickel, and zinc—the following metals: iron and steel, aluminium, tin, tungsten, molybdenum, tantalum, cobalt, bismuth, cadmium, titanium, zirconium, antimony, manganese, beryllium, chromium, germanium, vanadium, gallium, hafnium, indium, niobium, rhenium, and thallium, and their alloys.
Some googling led me to a summary from this book chapter:
Base metal is a wide-ranging term that refers either to metals inferior in value to those of gold and silver, or alternatively, to metals that are more chemically active than gold, silver, and the platinum metals (AGI, 1957). Accordingly, a review of base-metal mineralogy would encompass much of the world's metal production and geology. Usage of the term “base metal” in the minerals industry is rather loose, but a common application is to the nonferrous ore metals that include copper, lead, and zinc. Thus, e.g., Kesler (1994) grouped manganese, nickel, chromium, silicon, cobalt, molybdenum, vanadium, tungsten, niobium, and tellurium as ferroalloy metals, and copper, lead, zinc, and tin as base metals.
Not only do I agree with the comments by TheSimpliFire, S R Maiti, and wzkchem5, that there is likely no word for this in English; I'll also go as far as to say that even the German word you suggested does not precisely fit your description of being "non-ferrous and non-noble".
"Buntmetalle" just means "colorful metals".
The term is applied to non-ferrous metals excluding precious metals. It so happens that most metals that are usually considered "noble metals" are also usually considered "precious metals", but the latter two terms are not "rigorously defined". For example, page 40 of this 2006 book includes copper as a "noble metal", but copper is never considered a "precious metal".
Note that copper (sometimes considered a noble metal) is one of the metals in your list of buntmetalle!
Anyon's answer suggesting the term "base metals" is probably the closest you can get, but that term is also not "rigorously defined" (just like "noble metals", "precious metals", and even "buntmetalle"), so it often includes metals like rhenium which is one of the most precious metals out of all the precious metals.