Welcome. You are not correct in all your assumptions. However, you should take note of certain factors that mean you are not liable for the charge.
Firstly, no "fine" was issued and HCF referring to it as such indicates that they have breached the terms of your contract with them. A Parking Charge Notice (PCN) whether issued as a Notice to Driver (NtD) or as a postal Notice to Keeper (NtK) issued by an unregulated private parking company is not and can never be a "fine". It is nothing more than a speculative invoice for an alleged breach of contract by the driver.
GXS is not an authority that can issue "fines". They are a private company of ex-clamper thugs who operate at the fringe of lawfulness.
Please show us the relevant section of your lease agreement that deals with "fines" and differentiates between a statutory fine and an invoice from a private company. I think it is highly likely that HCF have moronically grouped together any form of charge as a fine, in breach of the terms of your vehicle lease.
As the PCN was from an unregulated private parking company, all HCF had to do was to comply with the requirements of PoFA Paragraph 13(2) and 13(3). If they have complied with that, then liability is transferred to you, the Hirer of the vehicle. GSX cannot later go back to HCF and make any more demands.
Was you have shown us is a copy of the NtK that was sent to HCF. How did you obtain that?
Once HCF have transferred liability to the Hirer, the creditor, GSX, is required to send you, the Hirer, a Notice to Hirer (NtH) together with copies of the documents that should have been provided to them by HCF. I will place money on the fact that GSX have not included copies of the documents as required under PoFA, in order to be able to hold you liable as the Hirer.
GSX (and HCF) have no idea who the driver is. Unless you, the Hirer, blab it to them, inadvertently or otherwise, they cannot hold you liable as the Hirer because they have not complied with all the requirements of PoFA.
There is no legal obligation on the Hirer (or the Keeper) to identify the driver to an unregulated private parking company. If GSX do not know the identity of the driver and the Hirer declines to identify the driver, they cannot do anything about this.
Unfortunately for you, or at least for HCF for having paid a speculative invoice from an unregulated private parking company, you and/or they are going to be out of pocket whilst GSX laugh all the way to the bank. You will have to check your lease agreement (please show us the relevant section) about admin fees. If HCF have breached the agreement by paying an invoice as opposed to a "fine" or a "penalty" that is issued under statutory law, then you will have to complain/sue HCF to repay the admin fee.
For now, you cannot deal with the PCN from GSX until you receive an NtH in your own name. When you do, you simply appeal with the following:
I am the Hirer of the vehicle and I dispute your 'parking charge'. I deny any liability or contractual agreement and I will be making a complaint about your predatory conduct to your client landowner.
As your Notice to Hirer (NtH) does not fully comply with ALL the requirements of Paragraphs 13 and 14 PoFA 2012, you are unable to hold the Hirer of the vehicle liable for the charge. Partial or even substantial compliance is not sufficient. There will be no admission as to who was driving and no inference or assumptions can be drawn. GSX has relied on contract law allegations of breach against the driver only.
The Hirer cannot be presumed or inferred to have been the driver, nor pursued under some twisted interpretation of the law of agency. Your NtH can only hold the driver liable. GSX have no hope at IAS, so you are urged to save us both a complete waste of time and cancel the PCN.
Please show us the relevant terms of your lease agreement so that we can confirm whether you are liable for the admin fee.
Edited to add: If HCF have stupidly paid the PCN, you will never receive any further correspondence from GSX (while the laugh all the way to the bank) and so you will not have any opportunity to appeal the PCN. HCF have unlawfully deprived you of your right to appeal the PCN.