Yes, you are correct, Liverpool St Station is in the City; its postal code is EC2.
As a Londoner until I was married at 22, living in Stoke Newington (N16), and often travelling to Shepherds Bush (W12), I thought of Liverpool St Station as being on the East side of London - after all, it is close to Bethnal Green and Shoreditch, which I thought of as really East End places.

Dirty Dick's was, as I remember, and I suppose is, located across the road from the SE entrance to the station, so we certainly are talking about the same place. The station was convenient for both Metropolitan and Central Line undergrounds (Can't call the Met a tube line, as it was mainly cut and cover). And, with stations in Shepherds Bush, each end of the green triangle in Shepherds Bush.

As a youngster, on occasion, I would go down to Shoreditch to buy a bottle of "Best Button Polish" which my cabinet maker father would use to finish off a piece of furniture. Although he was described as a Journeyman, he really was a craftsman.  I have a bedroom suite made by him for himself and his new wife back in about 1935. There is a huge drawer in a chest of drawers, full of blankets, which I used to be able to push in  by pressing on one corner of the drawer, with my little finger. I can't do it now, not because the drawer has stuck, but because my little finger is weak.

I and my new wife moved away from London, to begin a new life out of town in Harlow, Essex, eventually gravitating to our present home on the South Coast.