Small profile camera for recording a lecture (from a student's seat)
Since your camera will have a fixed position and you will be filming under stable lighting conditions, you don't need a fancy camera. An iPhone will do fine, the results will be satisfactory. Since a lecture isn't really about beautiful pictures anyway, I wouldn't put to much thought into it. 

However, if you want to invest some money in a fair camera, I would probably go for a Sony Camcorder, e.g. the Sony HDR-CX330. Those offer a good compromise between usability, recording quality and price. 

However, what is really important is the micro. If you use the micro of the camera, the audio will completely suck, no matter which camera you chose. Make sure to get a decent micro, rather than an expansive camera!

And of course, although you are probably aware of that: You need to ask your professor for permission ;)