I agree with Boditza. Try reading the codes. As far as I remember there are only 2 or 3 codes relevent to pre heating, as follows.
P 1694- Glow plug indicator lamp/ wiring fault
P 1694 - Has another explanation that I have forgotten, but it is associated with the Glow plug indicator lamp
P 1635 - Mal function in glow plug heating and control circuits.
The first is pretty self explanitory and the third can account for one or more glow plugs open circuit, wiring problems or glow plug control module fubared or just about any other pre heater malfunction other than the indicator bulb/ wiring.
It is not unknown to have all four glow plugs fubared and the car will still start quite easilly in warm weather. They are easy to test - pull off the caps (they have a release clip) and they should read about 1 ohm tested to earth in situ with a meter. If you have any that are higher resistance they are fubared.
Glow plug control module sits in the big black triagular plastic box on top of your battery.You are looking for a finned aluminium box about 9cm by 4cm right in the bottom of the triangular box. Part is common to several GM cars and is often held in stock by a decent dealer. GM# 55354141
IIRC you have changed the battery, so will have disturbed this module? It may be worth getting it out and checking all contacts are good, but see what the codes indicate first. When I had this problem I noticed that on distrurbing this plastic box, my starting problem disappeared, but returned a week later.
PS 2.2Dti has no lift pump