"For the purposes of this Regulation:
(a) ‘air carrier’ means an air transport undertaking with a valid
operating licence;
(b) ‘operating air carrier’ means an air carrier that performs or
intends to perform a flight under a contract with a
passenger or on behalf of another person, legal or natural,having a contract with that passenger"
For the record the "legal" entity in the last sentence refers in this case to the commercially and contractually responsible legal entity whose capacity has been / is oversold. The onus is on the "selling" entity, air carrier or otherwise, to maintain and manage the reservation capacity supply against demand (called, unsurprisingly, Revenue Management).
For as long as the actual Air Carrier fulfills their contractual obligation towards their principal there is no problem. Seldom does a (Charter) Air Carrier have the possibility to adjust physical capacity in an upwards direction, which is all well and good; however aircraft substitutions of an equal or lower physical capacity do on occasion occur, and in the latter case are covered by the terms of the Charter Contract with the Principal (read: Legal Entity).
Although it is the Air Carrier that might have to take the first blow from, as in this case, customers who have been denied boarding, they will in turn refer the matter to their Principal, who is responsible for managing all and any claims against the Contract of Transportation against which the travel arrangements have been sold.
It is not uncommon for customers to have take out, or automatically have insurance against this kind of unfortunate event.
It's never fun to be subjected to Denied Boarding and no amount of compensation in the world will at that moment be considered as "fair and reasonable" when watching the aircraft pushing back from the Gate.
All the best,
PB
(former 10-year veteran Customer Relations Manager, for a major Flag Carrier with a large cross on the aircraft tailfins that went bust in 2001)