I think if we refer to Alex's link earlier from Wikipedia, can clearly see that if you have a 320i, it only falls into two categories:
M52TUB20 which has a cubic capacity of 1991 CC (2000 cc )
OR
M54B22 which has a cubic capacity of 2171 CC (2200 cc)
M52 engine was used previously in E36 with single vanos. Whereas M52TU is the engine which has Technical Updates (hence the M52 --TU) which has double vanos and widely used in E46 320i.
The difference of M52TU and M54 is very minimal, limited to minimal ECU tuning and intake manifold.