Can anyone advise what my legal situation is here?
The situation is that they are alleging that you owe them £160 as liquidated damages for breach of a contract.
If they take you to court and can prove (on the balance of probabilities) that you as the driver (or vicariously as keeper if they can prove that they satisfied the requirements of the PoFA 2012) were in breach of such a contract and that such a penalty is commercially justified, then they may well be found to be entitled to recover the £100 (plus lodging fee, etc.).
The fact that private "Parking Charge Notices" so closely mimic local authority "Penalty Charge Notices" is largely irrelevant other than to give an indication of the integrity of the people you are dealing with.
The fact that you moved house without updating your V5C is your problem. Essentially it means that it is your fault that you didn't receive the initial threatograms and cannot readily establish whether they are entitled to hold you liable as keeper.
Contracts for parking are usually formed by performance - all things being equal (and they are not always equal) if there is clear signage ("there to be seen") setting out the terms of a contractual offer for parking, you are deemed to have accepted the offer by parking there.
The starting point of any contractual dispute is likely to involve the terms of the contract - what did the signs say?