How specific are the E53 diagnostics?
 

I'm wondering how specific the diagnostics are via the OBC port when trying to diagnose a fault?
Specifically I had a starting problem recently which the garage had a look at. They came back and said the diagnostics showed it was the fuel pump (in the fuel tank) which was faulty hence they replaced it. Since getting the car back the problem is still there so I'm concerned they replaced a part that didn't need replacing.
Do the diagnostics often point to a specific part or do they just generalise a problem as 'fuel problem' say?

http://www.obd-codes.com/trouble_codes/bmw/

at the above link you can find all BMW troubled codes. They usually don't show the exact part that has to be changed. a mechanic with experience can just guess which one is faulty

it points you to the general area of your problem. I'd double check to make sure they looked up the proper code. It is probably a electrical issue realted to the fuel pump...

It should be pretty specific. One month ago, the ABS, DSC and Brake light lit up, I plug BMW DIS system to diagnose, it showed the front left wheel speed sensor faulty, I think this diagonsis is very specific, isn't it? May be it is only for BMW DIS, not for OBD code reader.

Looking at the OBD codes in the link above they are quite specific for some things but not others. Not sure if this one "Diagnostic Module Tank Leakage (DM-TL) Pump Control Open Circuit" is related to the fuel pump.
Also does the BMW DIS generate different codes to the OBD?

The BMW DIS does not generate any code, but directly the fault. As I said, I diagosed my x for DSC, ABS and brake light light up, it showed 'the front left wheel speed sensor transmits implausible signal', something like that. Very specific diagosis.