No, because once you have exited at Kings Cross you will have reached "London Terminals". As your ticket will not have the Maltese Cross it won't work the gates on the Underground.
Provided the ticket isn't TOC-restricted or specifically issued to Kings Cross, it's still valid beyond Kings Cross, because of the Underground interavailability between Kings Cross and Old Street / Moorgate (though not intermediately).
Therefore
@greyman42 is perfectly entitled to exit at Kings Cross and use the Underground to Moorgate (even if their ticket might not automatically work the barriers).
The question is whether they are entitled to walk to Liverpool Street and exit there, and I cannot see any legal grounds for saying that this is not permitted; the Routeing Guide etc. limit the
trains you can take, not the station entrances or exits you can use.