What category to put these copper strikes in?

Official copper strike of an intended silver coin, released into circulation by accident or other by the Official Mint.

Should these coins be listed under Standard Curculation, Contemporary Counterfeit or something else?

In Norwegian coinage there are two known cases of this happening. Although the later types' circumstances are unknown, the first type is locally famous.

First type:

Mintmaster Henrik Christoffer Meyer got caught minting coins with a lot less silver contents than he was supposed to, and even some coins had no silver and were just plain copper strikes. He was sentenced in december 1727.

Known coins struck in copper from this period:

N#99206 1723 and 1724

N#99778 1727 HCM

Second type:

The circumstances for the minting of this coin is unknown, but have later been verified to be authentic by the Official Mint.

N#40310

https://auksjon.meyereek.no/Event/LotDetails/1477886/2-skilling-1804-Christian-VII-Avslag-i-kobber-Kv1

If the original silver coin was „standard circulating“ then the copper version should be the same, regardless of the circumstances. It is like today when the wrong planchet is used, it becomes an error but it is still a „standard circulating“ coin. Unless of course there is evidence of this being a trial strike, which from what you haven’t written, the answer is no and unknown.

If the coins were made for an official contract (just not to spec) and not like most often today at the side (off metal strikes and other intentional errors) they should be standard circulation (especially if they were still used ass currency).

The information that there was something afoot in the process of their production should be mentioned in the comments.

Like the reason why it the denomination was lowered is that their silver content was lower then it should have been because of that dude etc.