[The driver] parked the car in a private car park in Amersham controlled by Parking Eye on 27 July. She is not now sure whether she paid for two hours and exceeded it, or the limit of free parking was 2 hours and she exceeded that. The PCN does not make it clear either.
I would imagine it is the former - as the PCN states:
"The signage states that, as a paid car park, a Parking Charge is applicable if the motorist fails to make the appropriate tariff payment"
"By either not purchasing the appropriate parking time or by remaining at the car park for longer than permitted....the parking charge is now payable"
I take your point about the 'catch all' phrase being rather vague, although it is standard for ParkingEye charges, and I think it would be an uphill battle to convince POPLA or a judge that the charge is not owed on that basis alone. A counter argument is that the two options describe largely the same thing: if a motorist pays for 2 hours parking and stays for 2 and a half, then they have both not purchased the appropriate parking time
and stayed for longer than permitted.
I can obviously appeal citing vagueness of both the offence and the location but I guess Parking Eye will just turn that down automatically.
This might be a more applicable avenue. You're right that the location is rather vague - St Michael's Court isn't a particularly unique name for a location, and without a town or postcode it could refer to a number of locations. In order to hold you liable as the keeper (rather than holding the driver, who they do not know, liable), they need to comply with Schedule 4 of the Protection of Freedoms Act. 9(2)(a) of that Act says they must
"specify the vehicle, the relevant land on which it was parked and the period of parking to which the notice relates"
You could make an argument that they have failed to specify the land with their vague description. I'm not aware of any authority/cases that have provided a definition of what is required in order to satisfy this requirement. On issues that aren't clear cut, it can be hard to predict which way POPLA will go, so it depends somewhat on your attitude to risk. Some useful background reading from over on PePiPoo:
http://forums.pepipoo.com/index.php?showtopic=144900 - in that case (involving UKPC), the notice simply stated a road and a postcode. The road in question was rather long with multiple private car parks, so the appeal largely centred on a failure to specify the relevant land.
If you're going to go down this route, show us a draft of your proposed appeal before sending anything.