Yes, Halesia monticola is now a synonym for H. tetraptera var. monticola and is noted as such in our database.
Mountain silverbell (Halesia tetraptera var. monticola)
Halesia tetraptera Ellis has no synonyms, however, and Halesia carolina L. is still an accepted name, so they are two different species. Halesia carolina var. monticola was reclassified as a synonym for Halesia tetraptera var. monticola, so maybe that's what's being referred to in your source.
Wait a minute! I suppose there could be another explanation. I just noticed that you refer to H. carolina in your plant comment:
"The Carolina Silverbell has been assigned a newer official scientific name of (Halesia tetraptera) that means "four wings" in Greek, so (Halesia carolina) is the same species."
And to H. caroliniana in your response:
"on page 412 he shows Halesia tetraptera Ellis (formerly H. caroliniana, L.)"
Was "caroliniana" a typo or is that the name your source actually uses? If it's not a typing error, is it possible that there once was a Halesia caroliniana species that was reclassified so long ago that it's no longer listed in the main taxonomic databases?