A Parking Charge Notice (PCN) is just a term used to say that a "ticket" has been issued. What is important is how the PCN is issued. It can reissued either as a Notice to Driver (NtD) which is stuck on the windscreen or it can be issued as a postal Notice to Keeper (NtK).
The point is that you received an NtK and subsequently identified yourself, the Keeper, as the driver, when there is no legal obligation to do so. Until you identified as the driver, they had no idea who was driving, only that you are the registered keeper.
The driver is always liable for any charge. However, if the NtK is fully compliant with PoFA, they can shift the liability from the unknown driver to the known Keeper. If the NtK is not fully compliant with all the requirements of PoFA, and this one isn't, even if they pretend it is, then they cannot hold the Keeper liable. They are not allowed to presume or infer that the Keeper must also have been the driver.
So, this is why you should never identify the driver if you receive an NtK. The Keeper only needs to refer to the driver in the third person.
As you have identified as the driver, it matters not that you are the Keeper. They will hold you liable for the charge.
When they rejected the appeal, the provided a POPLA code which you can use to appeal to a supposedly "independent" appeals service, but that is not strictly true, as POPLA are funded by the very same companies that are trying to pursue you for the charge in the first place.
The POPLA code is valid for 33 days from the date of the appeal rejection. POPLA will not consider any mitigation. They only assess whether the PCN has been issued correctly. So, if you can persuade the POPLA assessor that the operator has failed to comply with any aspect of the law of the PPSCoP, then the PCN has been issued incorrectly.
In your case, you have to try and persuade the assessor that ParkingEye have issued the NtK incorrectly because an NtK cannot be issued the same day as the parking event. Also, you can appeal other points too, such as poor or missing signage, incorrect wording on the NtK or missing periods pf parking being noted.
In your case, there is no evidence that the vehicle was parked for longer than the minimum consideration period (5 minutes). The driver could simply have parked, gone to seek out a sign with the terms and conditions on it, read those terms and conditions and decided not to accept them and left. No contract could have been formed and this is simply a dispute over a contractual matter.
Whilst it is a weak argument in your case because you have already blabbed about why you were there and for how long, it could still be argued that without evidencing a "period of parking", it cannot be proved that a contract was formed with the driver.
It's a technicality, but a useful one, as long as you don't go throwing away all your cards by giving the operator all the evidence they need to hold you liable.
Have a search of the forum for other POPLA appeals to see how they are laid out and argued.