Finding the negative feedback loop in the amplifier circuit
Why is my way of finding a loop wrong and theirs right? Isn't the loop always from Vout to Vin or is there something I'm not understanding here?


No, the loop isn't always from \$V_{out}\$ to \$V_{in}\$.  A loop happens if you can trace the signal path in a loop.  In this case the loop that your prof is talking about is the common-emitter stage around Q1, which is applied to the base of Q2, where the emitter of Q2 follows Q2's base voltage.  Then that emitter voltage is applied to R5, which feeds back to the base of Q1.  Yes, \$V_{out}\$ is being picked off of Q2's collector -- but that's not part of the loop in question.