What records should a CFI keep to prove they took due care in the quality of their instruction?
Have a well defined syllabus, and then a daily activity sheet where you write down the lesson you did, the time it took, and how the student performed. If anything peculiar happens, make a note. 

Have a well-written lesson binder too. Even if you don't need to use the binder to teach a lesson (like steep turns, that's not really much of a ground lesson), have it in there anyway to show that you know what you're talking about. If you have a page that only says "Teach steep turns" and a lawyer gets a hold of it, there's nothing to prove that you actually know how to teach steep turns.

AS for legally required records:


You must sign the logbook of anyone you have given flight or ground training.
You must maintain a record of the following:

The date and name of each person whose logbook/student pilot certificate you endorse for solo flight privileges (I keep a record of every endorsement).
The date and name of each person you endorse for a knowledge/practical test, and the kind, date, and result of the test.

You must keep the records for at least 3 years.