DF: I'll try to answer your question.
I don't know of any official definition of "funeral gun", but here is my take on it FWIW.
When I hear/see reference to a funeral gun I expect to see: a Brit SxS, sidelock, "name' maker, higher grade gun but with no/very little engraving.
As better quality Brit sidelocks traditionally had goodly amounts of engraving, the lack of engraving on these guns was UNUSUAL and left a relatively large expanse of vacant, blue emptiness on the sidelocks. This "untoward" unadorned expanse of blue/black finish brought to mind the standard "funeral' colour - hence "funeral gun".
I would not argue with anyone who wanted to include in my definition, a HIGH grade American gun lacking the expected/grade standard engraving that "lettered" as being ordered "funeral".
A standard factory production unengraved gun would not be considered a funeral gun, IMHO.
As this is a somewhat nebulous term, I expect others have their versions of truth also - but this is MY version.