The Jargon File (version 4.4.7, 29 Dec 2003):

vanity domain
 n.

    [common; from ?vanity plate? as in car license plate] An Internet domain,
    particularly in the .com or .org top-level domains, apparently created for
    no reason other than boosting the creator's ego.


The Free On-line Dictionary of Computing (30 December 2018):

vanity domain

    A domain you register for the sole purpose of
   having your own domain so you can have an easily remembered
   URL and e-mail address.  The domain is usually served
   (often vhosted) off someone else's machines.

   This is as opposed to a domain you register because you have
   machines of your own which are already on the Internet and
   which you want to make addressable via something other than
   dot addresses.

   Whereas vanity domains were almost unheard-of in 1980s, since
   the invention and popularisation of the Web in the mid-1990s
   and the desire for URLs which consist only of memorable
   domain names (e.g., "http://pbs.org") for everything from
   movies to car wax, vanity domains have come to be the rule
   instead of the exception.

   (1997-09-11)