The cross London part of a ticket is purely for that, crossing London to get the next station your train is going from.
So I can go from any
TfL» station to any TfL station? Or does it have to be Paddington - Liverpool Street, or can it be Lancaster Gate - Liverpool Street or even Waterloo - Tower Hill for example?
No. You can make a single Tube journey from the London terminus you arrive in to the London terminus you depart from.
At least that is the general rule (which you should stick to until informed differently). Hopefully someone will know if there are limited and reasonable exceptions to it when you are talking about Lancaster gate or Euston Square