Well, first off, their letter is correct in stating there is more than one machine, because here it is: -
https://maps.app.goo.gl/mFXofFz64Qafv94Z8You can see the car park name is the same as the one you took photos of. So what appears to be two separate car parks is actually a single named car park.
As for the sign, I am puzzled why you took the letter as meaning payment was not needed, because it doesn't say that.
However, their letter is quite disingenuous, because any normal visitor would assume there were two separate car parks. Just covering up a ticket machine without giving any notice of where another might be located is also dreadfully bad practice, bearing in mind the other machine is not visible from the one you went to.
The main point now, is whether you feel sufficiently aggrieved enough to stand your ground and take the matter all the way to adjudication at the Traffic Penalty Tribunal (TPT). This would mean foregoing the discount, waiting for the Notice to Owner, (sent to the name and address on the V5 Registration Certificate for the car), submitting a formal representation, getting rejected, then registering an appeal at the TPT. Appeals at the TPT are mostly by phone, but there is now a remote video facility offered.
https://www.trafficpenaltytribunal.gov.uk/My own opinion is that going to adjudication would be something of a gamble, but others may differ so wait a bit and see what they say.