If you go down to the local lumberyard today and ask for "red oak", you'll be shown a selection of what passes in the trade for "red oak"--they don't break it down by species (because, really, who cares as long as the woods generally work and finish the same?). Similarly, asking for "white oak" will get you any of the "white oaks". If you're dealing with a local sawmill, they may identify the woods by local names and not even know which are white oaks and which are red oaks. (So is "burr oak" a white oak or a red oak, and how do we know the oak was correctly identified in the first place?). (For the curious, "burr oak" is technically Quercus macrocarpa, and a white oak.) I deal a lot with a few small sawmills, and won't buy oak without doing some testing so I know what I'm getting--because for some uses it does matter. But much of the time, it doesn't matter today.
Species names didn't matter a lot back in the day, either. You might place an order with an importer or timber company for "Brazilian rosewood" and get any of the Dahlbergia species from (roughly) Central America; more likely, you went down to the dock or warehouse and picked out the wood (and as long as it matched, and the working characteristics were the same, who cared about species?).
We have names like "box" and "camwood" and "barrwood" and "coco" and (according to Denig and pretty much no one else) "logwood" described in fur trade records as being used for knife handles. Records of buyers in Europe identified or specified "red wood", "box", "beech", "Campeche", "coco", "vermillion", and a couple others. These are common names or trade names (much like "red oak" or "white oak" is used in the trade today). In a few cases, we can get multiple names ("red wood" and "coco", "red wood" and "Campeche") in records that seem to be referring to the same order or orders. If we want to know what specific woods were being used, we have to do some work.
If we study old records (ships bills of lading, taxes, timber dealers, cabinet shops, cloth trade, etc), and examining/identifying original knives and furniture where the woods were identified at time of purchase, we get the following:
"Barrwood": based on records and a few extant pieces of furniture, this one most-commonly (but see also "Camwood" below) appears to be Andaman padauk Pterocarpus dalbergioides. Originally from the Andaman Islands and India. I've also seen this wood called "vermillion" or "vermillion wood" in old records. Ages to a reddish or purple-ish brown. Hard to find today (not enough data for a CITES listing, but possibly Threatened or Endangered). The common substitute today is African padauk, but the African species is more brown, lacking the reddish tone as it ages. For our purposes, African padauk is probably not a good substitute due to the color difference as the red was apparently valued.
"Beech" probably refers to Fagus sylvatica. This was and is a common utility wood. European beech does differ slightly in appearance from American beeches, but as it is easily available there is probably no need to seek substitutes.
"Box" is any of several species of boxwood (Buxus species) sourced from Europe and the Caucasus. The species from the Americas, and from Africa, are more-closely related to each other, with the Europe/Asia species more isolate. The most-common species sourced in Europe was probably Buxus semipervirens, but there arguments amongst some taxonomists as to whether or not the species from Turkey/Iran/parts of the Mediterranean are synonyms of B. semipervirens or not. In old records I've seen it referred to as "box", "boxwood", "Turkish boxwood". Box was used for ornamental turnings, engravings (block prints), musical instruments, etc, and the best could be rather expensive at times. I've long suspected that the wood used for cheap knife handles was sourced from offcuts and other wood that wasn't suitable for other uses, but have never found records to determine this. European boxwood is still available in the lumber trade; Castello boxwood is probably not a good substitute due to color and grain differences.
"Campeche" in old records appears to refer to two very different woods, and I suspect this is where Denig got turned around. It usually referred to mohoganys (Swietenia species) from Mexico/Central America, but I've also seen a couple mentions of "Campeche wood" in what were clearly references to logwood. Many of the loggers cut logwood as a primary product, and mohogany as a secondary product. My suspicion is that Denig got the two confused. I've seen one original pre-1890 butcher knife with what appeared to be a handle made from mohogany. It was a common wood in the trade so it wouldn't be a surprise if someone tried it for knife scales, particularly since it ages to a reddish brown. Available today in the timber trade.
"Camwood" is Baphia nitida. (It also appears in old records as "barwood"--possibly a source of confusion. Lord knows we don't have enough confusion on this topic.....) This is another red wood, often used as a source for dye but sold in the period for small work and knife handles as well. I've found this one a couple times in the specialty timber trade.
"Coco" most probably referred to cocobolo; more likely it was a trade name for any of the Central American Dahlbergia species. Easily found in the timber trade today.
"Logwood" was listed by Denig, but as noted above under "Campeche" I think this was due to confusion on his part. Logwood was valued as a dyewood, and since any piece of logwood could be used to make dye, there were few other uses in the trade back then. Available in the trade today.
"Red wood" could have been any of the reddish woods. From the context of many of the old records, I'd go with Andaman padauk or barrwood.
"Vermillion" seems to have been used in the trade to refer to Andaman padauk. I've not been able to connect the name to any other wood.