The above is all correct. Their Notice to Keeper (NtK) is not fully compliant with all the requirements of PoFA and therefore the Keeper cannot be liable for the charge. They have no idea of the drivers identity unless you give it away, inadvertently or otherwise. There is no legal obligation on the known Keeper (you) to identify the unknown driver (either you or someone else) to an unregulated private parking company.
In order to be able to transfer liability from the unknown driver to the known keeper, they must comply fully with all the requirements of PoFA, in both their NtK and their signage. As stated, they have not complied with POFA Paragraph 9(2)(a) in their notice which requires them to state the period of parking. This has been established in persuasive appellate case law that a period of parking cannot simply be stated as "the period prior to XX:XX". It must be a definitive period, not necessarily the whole period. As there has to be a consideration period of at least 5 minutes according to the ATA Code of Practice, then the period recored should be at least that long. In this case, It is not!
In the appeal provided, you can add that you are the Keeper and you can confirm that you also were not the driver. The only issue is that they re going to reject any appeal (no money in it for them if they don't) and they are members of the IPC which means that their secondary appeals service, the IAS, is just a kangaroo court set up to only protect their members interests and not those of the appellant.
What I can assure you of, is that if this is eventually escalated to a county court claim, you will to be paying a penny to HX. Any claim is very easily defended and the most likely outcome will be either a strike out or a discontinuation. Even in the remote chance it ever did make it all the way to a hearing before a district judge, you would win anyway.
These ex-clamper thugs abuse the system and their M.O. is to hope that you are low-hanging fruit on the gullible tree who will eventually capitulate and pay up out of ignorance and fear. We know how to deal with them and anyone following our advice will not pay a penny to these rogue operators.