Public Transport
Under the UK rules there were two Certificates of airworthiness. 'Public Transport' and 'Private'.
Flying Instructing has never been public transport but 'Ariel Work'. To undertake Ariel Work passengers could not be carried. Crew could be of course. There were many arguments as to what defined 'crew' as apposed to 'passengers'. A student pilot is clearly a member of the crew.
Ariel work could not be conducted in Private Catergory aircraft. To overcome this the CAA issued some specific 'exemptions'. Notably the sole owner of an aircraft could train in their 'private' catergory aeroplane with some provisos. But if you couldn't comply with the provisos' then you couldn't train.
Under the EASA regulations, now in force, the two Certificates are 'Standard' and 'Public Transport'. You only require to hold a Public Transport Certificate for the carriage of paying passengers. So the old understandings are defunct.