This list is extracted from a wood-name list of some 26,000 species and 140,000 common names that I have accumulated over the years.
Some notes/caveats about the species listed here and the common names associated with them.
I have removed a few species from the genus Ulmus and some common names. The common names I removed were all, as far as I could tell, names which are only used in countries other than the USA. The species that were removed were ones which, as far as I could tell with a quick look, do not grow in the USA and given how common elm is in North America there seems to be little reason to import them so I assume they are not present for the American woodworker.
Common names are limited to Enlish language ones that have the word "elm" in them
I do NOT believe that this list is comprehensive, even given the caveats above, but it will give a strong sense of the number of elm species that grow in North America and the common names that are associated with them.
Apologies for the inappropriate lack of capitalization on the Genus name throughout and on the proper names such as "california".