Actually, British aircraft also had internal factory-only model designations (as did many US aircraft, see the B-25 etc).
Yes, but these were manufacturer designations, NOT Air Ministry designations. The Air Ministry specifications to which aircraft were built were quoted in correspondence, whether the aircraft was originally a private venture or not. The Lancaster was not developed in a vacuum, there was a constant stream of communications between the company, the MAP and the representatives of the Air Ministry.
In November 1940, Avro was instructed by the Air Ministry to stop referring to the four engine Manchester as the Lancaster and to use the name Manchester III in all communications.
The name 'Lancaster' was approved by the Air Ministry in a letter to Dobson, at Avro, shortly before the first prototype (BT308) left the factory in January 1941.
All service aircraft had an Air Ministry approved name, by which they were known in all correspondence and other documentation. The factory designations accorded by the manufacturers were utterly irrelevant.
The Spitfire was designated the Type 300 by Supermarine, until it got a name. After K5054 received its Air Ministry Certificate of Design on 6 March 1936 the company wrote to the Air Ministry (on 10 March), requesting a name.
"We refer to our conversation of this morning with A.E.Slater. Would you be good enough to reserve the name 'Spitfire' for our fighter to Specn: F 7/30 modified."
It was Slater who replied, following protocol.
"It has been noted for consideration when, and if, an order is placed for this aircraft."
It wasn't until 10 June that the Air Ministry wrote again to Supermarine informing the company that the name was now approved for
"F 7/30 (F37/34 modified)"
From that moment the Spitfire became officially the Spitfire. All future correspondence would refer to the type as Spitfire (except the Aussies who called it by the code word 'Capstan').
Notice anything missing from the correspondence? The internal company designation 'Type 300'. This was only relevant to Supermarine and its various offices.