I don't see your point? They still wouldn't "terminate" at Liverpool Lime Street!
On the way from Chester to Liverpool Lime Street, the announcement says: "This is the xx.xx Merseyrail service to Liverpool Lime Street." The front of the train says "Liverpool Lime Street."
When the train arrives at Liverpool Lime Street, the announcement says: "We have now arrived at Liverpool Lime Street. This train terminates here." Then the driver immediate changes the front of the train to "Chester," and the announcement says: "Welcome aboard this xx.xx Merseyrail service to Chester."
So really they are separate - but consecutive - services. It prevents people using the "this is a direct train" excuse, helps to avoid confusion, and still allows passengers travelling from Moorfields and stations prior to take direct trains to Liverpool Central and stations beyond.
Then again, I have seen a fatal flaw in this reasoning, which is that people won't know which station to train continues to after Liverpool Lime Street until it arrives at Liverpool Lime Street!

But I'm sure there would be ways around that...