Yes, it was a separate offense (so your mum shouldn't do it again) but this doesn't matter as far as this PCN is concerned. The recipient of a PCN only has to defend themselves against the specific allegation made on the PCN. The authority only has one go at getting this right, they can't go back and issue a second PCN if they messed up.
How you should proceed for now is as suggested, a very simple one line response pointing out that their own photo shows the vehicle was not on the carriageway and therefore the PCN must be cancelled.
Everything else you've said is potentially useful later on if the Council don't fold immediately, but at the moment you're dealing with junior level council staff who probably have to hit a tough target for number of PCN responses processed. Your objective is to get them to realise the traffic warden messed up, so they press the button that spits out a standard form 'on this occasion we've decided to let you off' letter. If they don't, then you can go in all guns blazing at the adjudicator stage, armed with a ridiculous rejection of an obviously winning argument.