I believe you are being asked not if it was a mobile or paper ticket, but was it an Advance ticket or an Off Peak or Anytime ticket? Also was there any "routing on the ticket? E.g. Any Permitted or EMT Only or something similar.
I would say in this case, if your account is true, upon the EMT train being cancelled you received permission from the Authorised Person to travel on the Cross Country train you took. If needed, in giving permission to travel on the next service after the EMT being cancelled, then it is for member of staff to ask what type of ticket you hold/or to see it before giving such permission.
I also do not agree that ticket acceptance only applies during major disruption. Sometimes it is better for all concerned, staff and railway staff, following severe delay/cancellation of a train to authorise passengers to travel on the next train regardless of operator. It keeps people moving and gets rid of a train load of passengers for the staff to have to deal with otherwise. Best outcome for all. It is for EMT to sort out with XC though. However, if you've had permission that it is not your problem if the railway have messed it up.
The attitude and approach by the XC Train Manager, if what you are saying is true, is disgusting. I do not like the Police being used as a threat to try and get people to comply with what they want. It is a form of bullying and it is not some ace card that can be pulled on a passenger by certain employees when they see fit or don't get their own way. The Police are not the railway's private "rent a thug" and they are there to serve anyone. The XC staff member clearly was wasting Police time and it was not a Police matter. If you had permission to travel then it is certainly not a Police matter. How long were you delayed in terms of both your arrival into Derby following your cancelled train and also, upon arrival, how long was spent dealing with the Police?
There really does seem to be a very militant approach being taken by XC train managers across the country. I wonder why this is? This would normally not be advisable for a lone worker working in a customer facing role.