Well, I’ll give you my opinion. Ray might disagree. None of those cartridges are “wildcats.” Just because an individual who is not an employee of the firearms industry, necessarily, designs a cartridge doesn’t make it automatically a wildcat. All of the cartridges you mention are in properly headstamped, factory-made brass.
One could argue that they are proprietary cartridges, and I would agree with that classification for most of them.
These are not even like the case of, say, the .22-250, where the cartridge was a wildcat for some years before it became “legitimatized” by factory production. As far as I know, in each case you mentioned, the first production of the cartridges was with properly headstamped brass made by a recognized manufacturer. I am not talking about prototype loads, but rather cases and ammunition presented to the public. That is certainly the case with the Hornady 38 TJ, AP 38 Super RL, and the Cowboy 45 Spec. I am not 100% sure about the others. I suppose there is a chance that they appeared around as wildcats first. Regardless, to me, they lost that status the minute properly headstamped brass from recognized manufacturers became available.
If they were only made for one individual for his own use, like the .257 Condor seems to have been, or one individual to sell exclusively, like the .45 Cowboy Special, that actually is the purest use of the term “Proprietary.” Wildcats are not generally “proprietary,” in that once the dimensions are released, anyone can have dies made and make the cartridge for sale (if complying with whatever licensing laws would apply in their region), or for their own use. In the case of a round like the .45 Cowboy Special, the brass maker will not sell the cases to anyone except the party that contracted for their manufacture, and thereby has the exclusive sales rights. Even then, he is probably not protected, unless there is some patentable feature about the round, from another ammo company making the rounds, even if under another name if the name is copyrighted.
Again, just my opinion. Other opinions welcome, but please use a lightweight truncheon when beating me about the head and shoulders for my comments.