Then the wards should be saying now that the Defense Professor is dead?

I hadn't thought of that. Some possible explanations:

1) Quirrel didn't actually use that trick.

2) The wards don't report the deaths of faculty, only students.

3) As long as any part of "Defense Professor" has is alive, the wards don't register its death.

4) The wards did register the Defense Professor's death, but either

4a) Dumbledore didn't notice the extra death report, since he was preoccupied with a student's death, and didn't look it up afterwards

Or

4b) Dumbledore knows about it, and (correctly) assumes that the same glitch that attributed the troll's actions to the Defense Professor attributed the troll's death to the Defense Professor.