Audi a4 B6 2001 diesel 1,9 96kW

It's difficult to start my audi.. I have to crank about 3-5 seconds till the engine starts. No matter what's the temperature outdoor (I have this problem since summer). Sometimes the engine starts normal (few cranks) but it is about once in 20 attempts. And sometimes, but really rarely the engine starts normal almost all day. This problem is since I brought the car (3 month). Interesting, but if the engine is running and I turn it off, wait about 0 -10 seconds or little more - then if I try to start, it always starts good.

I changed all filters, fixed starter (because in some auto repair guys claimed - that's the problem. I know the battery is good, because with this battery other car starts normal, and I tried to use powerful almost new battery, the starter was cranking very fast, but engine started with same delay. Some service guys performed diagnostic, but everything was good, just one nozzle's parameters were a little different, but they said it's in normal range. In fuel system is normal pressure, so it's not fuel system problem.
Also car exhaust black smokes.

Please could somebody tell me what's the problem???