Dispensary Districts were created by the Medical Charities Act 1851. When state registration of births and deaths started in 1864, each Registrar's District was coterminous with a Dispensary District, and each Superintendent Registrar's District was coterminous with a Poor Law District (containing a number of Dispensary Districts).
In urban areas, Dispensary Districts (and therefore Registration Districts) were often referred to by number, presumably in order to avoid the confusion of differing usages of names.
Over time, boundaries of Dispensary Districts and Registration Districts have changed - in all formal changes which I have found, the boundaries are defined my mereings (verbal descriptions), not by maps.
My personal experience (when my father died in April 2006) is that Registration Districts in Northern Ireland do not to this very day have maps to define their area. How on earth do they cope, you might ask - you will hardly believe it, but the Registrar rang the binmen in her District Council, and asked them if they collected from my father's house!
Adrian