Does taxiing in London City (LCY) happen on the runway?
To answer the title question, yes.

There are nice timelapse videos for EGLC/LCY on YouTube, if you want to see it in action. (Example.)

As for the actual operation at the airport, below is the official chart and textual data via the UK AIP:


  6 USE OF RUNWAYS
  
  b) Minimum Runway Occupancy Time - Arriving Aircraft.
  
  
  Pilots are reminded that prompt exit from the runway enables ATC to apply minimum spacing on final approach that will achieve maximum runway utilisation and will minimise the occurrence of 'go-arounds'.
  When landing on Runway 09 pilots should commence back-track as soon as practicable and exit via Hold Echo unless otherwise instructed by ATC. Any aircraft that continues landing roll beyond Hold Kilo may infringe the ILS critical area.
  When landing on Runway 27 A318 and CS100 pilots should plan to exit at Hold Delta (after a back-track if required). A318 and CS100 aircraft are not permitted to exit via Holds Charlie, Bravo, or Alpha. All other aircraft types may use any Hold to exit the runway except Hold Echo which shall only be used when specifically instructed by ATC. Pilots should be aware that use of Hold Alpha will increase Runway Occupancy Time.
  Pilots expecting to use the full runway length to stop (e.g. due aircraft weight/meteorological conditions) are requested to inform Thames Radar on first contact.
  


(Bold emphasis mine.)

The short version (runway 9): backtrack [on the runway] to vacate as soon as possible (example from cockpit) using the specified exits even if other exits may seem to be available, and if you need full runway length, notify the approach controller (Thames Radar).

(If the pilot does need full length, the approach controller needs to know to add extra spacing for the plane behind.)

For the exits mentioned in the text, reference the chart below: