No, for BA flights there are just two rates of APD:

Reduced rate for the lowest class cabin in the aircraft (also has to have seat pitch of less than 40 inches, so Baby Bus doesn't get into this category!), and
Standard rate for all other seats.

There is a higher third rate of APD, but this only affects aircraft of more than 20 tonnes with fewer than 19 seats: i.e. corporate jets.

https://www.gov.uk/guidance/rates-an...uty#rate-bands