When I worked as a driver for a haulage company that specialised in chilled transport I noticed that meat products usually carried a label indicating the species from whence it came. Beef - Bovine, Lamb - Ovine etc.
This is because such things are mandated by the EU and they needed a system that would be understood in all member countries.
Sadly, the labels were not always accurate and it was later found that some ‘bovine’ products should have been labelled ‘bovine/equine’.
My first WAG was that it might be the difference between whether it’s a Greek goat or a Latinate goat. But it appears more likely that it’s based on whether the goat is domesticated (hircine) or not (caprine).
Pachyderm (“thick skin”) is an outdated term that grouped certain large animals that are now known not to be closely related. “Elephantine” is an adjective but it refers to size rather than to the animal itself. I’m not aware of adjectives that describe the other animals.
On top of which, to the average person in the street, “pachyderm” is probably synonymous with “elephant”. That said, “pachydermous” has a nice ring to it. So nice that it’s in the dictionary, and means what you are after.
Little known fact: horses and pigs were once included among the Pachydermata (horses because of their relationship to rhinos and tapirs, pigs because they were thought to be related to hippos). Basically it included all hoofed animals that were not ruminants.
These names change. Taxonomy and Systematics are different, but overlap a great deal. They both came into being long after the Romans. Now we have three kingdoms. Plant, animal, and fungus. The snow leopard may get its own genus, uncia, since it only has 5 skull bones, unlike the 6 other members of panthera have. They still don’t know what to do with protozoans, and virus still lists as a biological agent. None of this is set in stone according to my friends at the natural history museum.
Three? I first learned five in the 1980’s, adding Protista and Monera to that list. But the number has been all over the map, from two to dozens. The problems of monophyly (increasingly seen as of overriding importance in systematic biology) has kinda put the kibosh on traditional kingdoms, especially groupings like the Protista. I have no idea if the concept is even being taught in college-level biology anymore, though I wouldn’t be surprised if it is lingering in High Schools.
Yes, now we now have three Domains above Kingdom, Archaea, Bacteria, and Eucarya. Protista are known to be completely paraphyletic compared to multicelluar life, some being closer to animals, some to green plants, and some to fungi. So really the Kingdom level would no longer be used by anyone using cladistics (or if you did, as Tamerlane says, you would need dozens of Kingdom-level categories, not just three or five.)