Whilst your mitigating circumstances are understandable, unfortunately you are not dealing with anyone who has an ounce of care about that. The PPSCoP allows for an additional minimum of 10 minutes "grace period" and you have exceeded that by 9 minutes.
There is a minimum 5 minute "consideration" period for you to park, find a sign, read it and decide whether you agree to the terms and conditions and leave, but once you have remained longer than that, you are deemed to have accepted those terms and the consideration period is not extra time that can be added to the grace period.
So, technically, the driver has breached the contractual terms of parking. However, you are dealing with (not so) Smart Parking. There is every likelyhood that their Notice to Keeper (NtK) is not fully compliant with ALL the requirements of PoFA and so cannot transfer liability from the unknown (to them) driver to the known Keeper.
Unfortunately, the way you have described the situations you have made it very obvious that you, the person that received the NtK, the known Keeper, was also the unknown Keeper, because you have blabbed it, albeit unintentionally, because, as the Keeper, you have not referred to the driver in the third person.
There is no legal obligation on the Keeper to identify the driver to an unregulated private parking company. If the NtK is not fully compliant with PoFA, then they cannot hold the Keeper liable. If they don't know who the driver is, then they have nowhere to go with this.
There is also a high likelihood that (not so) Smart Parking do not have a valid agreement flowing from the landowner thugs permits them to operate and issue PCNs at the location.
So, until you show us the NtK with only your personal details and the PCN number redacted, but leaving ALL dates and times visible, there is not much more we can advise as the moment/ Suffice it to say, if you do follow the advice, you won't be paying a penny to (not so) Smart Parking.