IMHO it is bad form to include legal argument in a witness statement. However everyone, particularly the parking companies and their lawyers, does that, so don't expect any criticism on that point by anyone other than me. IMHO (again) the proper way to clarify the defence submitted is though a skeleton argument.
Generally, once your defence has been submitted, you cannot amend your defence or add new points to it (as opposed to explaining or expanding upon the legal points already submitted) without the permission of the court. Whilst the rules in the small claims track are often applied more loosely (it is intended largely for litigants in person, and the cost of legal representation is generally not recoverable unless the other party is deemed to have behaved particularly unreasonably), it would seem prudent to nominally request permission in whatever form (or title) or legal argument you are submitting , and to explain that as stated in your original defence, it was not practicable to advance a meaningful defence as the PoC were inadequate and did not disclose any cause of action. Now that C has clarified their position in that the claim is purportedly brought under contract, despite the facts of the case seeming not to support this, you are now have a slightly less nebulous flawed claim to defend.
For a contract to be formed there has to be an offer of valuable consideration. That does not have to be the right to park. The issue is that there was no offer of any valuable consideration, communicated by the sign, or othwerwise and therefore no contract could be formed by any deemed acceptance. The offer of consideration that one might expect from a sign communicating an offer capable of forming a contract by acceptance by performance would be the right to park, but the fact that there was no offer of a right to park is not what defeats any claim of contractual liability, it is the lack of *any* valuable consideration being offered to the driver.
This is your case, not ours, but I would suggest that your understanding of the phrase "front and centre" differs from mine and DWMB2's. What you do with that information has no financial effect on myself or DWMB2.
As regards the "rules" of the small claims track. Perhaps the most important is "never ambush a District Judge" (regardless of whether the court has required or even invited disclosure of skeleton arguments (and regardless of whether the author of such skeleton arguments has written "Skeleton Argument" or "Witless Statement" at the top of the page.