By 1839, on the eve of the Great Landslip, it had its present name. Would the earlier formation, formed by the 1775 landslip, also be known as the Undercliff?
Donald Campbell, Exploring the Undercliffs
Elaine Franks, The Undercliff
ed. Deirdre de la Faye, Jane Austen's Letters
Patricia Pierce, Jurassic Mary
British Geological Survey website
Paintings by Copplestone Warre Bampfyld (such a resoundingly eighteenth-century name.)
This website: http://www.southampton.ac.uk/~imw/Lyme-Regis-Westward.htm
A little detail indeed, but I don't want to commit anachronism!