How to identify the circuit has negative feedback or positive feedback?
It will work with nearly all circuits, not just op-amp circuits.  


And I'm only saying "nearly all" because there's always exceptions -- just assume that it always works.
It can get tricky if you break it at a point where the resulting circuit's output impedance is close to the resulting circuit's input impedance.  Then you have to account for how that "input" loads that "output".
Note, too that for some circuits and for some break points, you may want to consider the feedback to be a current, not a voltage -- there are some transistor amplifiers like this.

Your instructor was probably breaking the circuit at the op-amp output because the output impedance is low, and it "looks into" a relatively high input impedance.  As an exercise, you can always take some of his examples and try breaking the circuit at some other point (the next most convenient point in an op-amp circuit would be at the \$V_-\$ pin).  Work out the math, and see if you get an answer that's the same, or substantially so.