No, BA has many different internal codes for their A320 family fleet, including:
A318
18A: A318 - EUNx
A319
19A: A319 - EUxx (new shorthaul cabin)
19B: A319 - EUOC, EUPG/T (old shorthaul cabin)
M9A: A319 - DBCA-F/J/K
M9C: A319 - DBCG-I
A320
10D+: A320 - GATx
20A: A320 - EUxx, TTOx (new shorthaul cabin)
20E+: A320 - GATx
M0A: A320 - MIDx
M0C: A320 - MEDK
A321
21B: A321 - EUXx (new shorthaul cabin)
M1B: A321 - MEDx