Keith Snyder proposes a new definition of "vanity press" that reflects the way self-publishing has evolved since the advent of print-on-demand technology:
We need to stop telling people a vanity press is a company that charges fees, and start telling them a vanity press is a
company that makes the bulk of its money from a very large number of
very small print runs that it sells mainly to people the authors know.
That’s a 21st-century vanity press.
He may be on to something here, though the definition may need a little refining to fully cover scams like PublishAmerica and their ilk.
(Thanks to Paul Guyot for the heads-up).