Transfer Function vs Frequency Response
Was my prof wrong? 


No. You're right, you can find multiple systems with the same frequency response, or the same transfer function.

But if you're using something as a (linear, which is often implied) filter, then all you care about is the frequency response (otherwise, you'd probably not be calling the thing "filter"). So, when I define a filter, it's usually sufficient to specify $H(\omega)$.