I worked in the TV industry for over 45 years. The camera in the $25 cell phone that I used before the change over to 5G (my old CHEAP cell phone) is better in terms of the image quality than any professional camera the stations I worked for used in that entire period. All it lacks is the professional bells and whistles (and weight and size).
I would estimate that at least 80% of the videos on the internet today are made with one or more cell phone cameras.
I would suggest a good/your best cell phone on a tripod at the back of the classroom. Use a wide shot if you plan to move around the room so it catches wherever you plan to go. Or to catch any student questions or comments.
A second camera/cell phone can be located near the front to provide a tight shot (speaker only) when he/you are at the primary location in the lecture.
The use of two cameras/cell phones will also allow you to record two audio tracks. One can record the professor's (your) microphone while the other can use the built-in microphone to record the room for questions/comments from the students.
The video and audio tracks from the two can be easily combined with modern inexpensive or free editing software (that's apps for the great unwashed).
Now, here's the trick to cheap tracking. Since most of the videos on the internet today are in a lower version of high resolution (1080P or lower), if the camera at the rear of the room is set for a higher resolution, like 4K video, then the editing software can zoom in to a tighter shot AFTER the footage has been recorded. It catches it all and you pick the tight shots later. Modern, high end cell phones or video cameras can record the video in these higher resolutions. A 4K video image (iPhone or many others) has about 16 times the number of pixels as a 1080 image and this allows a 4X zoom while keeping 1080P resolution or better. Beyond that, depending on just how tight you zoom in, this many not be a full 1080P resolution, but with good editing software very few viewers will ever see or know the difference. This will require some editing time, but it is probably the least expensive way in terms of equipment. And you will not need a camera man to control the camera during the lecture. You do the editing after the fact, at your leisure.
I am presently collecting the means of shooting some internet videos. I plan to use my newest cell phone, a medium priced Android model, as the primary camera and my old, 3G phone as a second camera or as a viewfinder for the primary camera. I also have two relatively inexpensive digital cameras which are primarily intended for still photography but both will take video footage as well. Virtually every digital camera out there is also a video camera. One or both of them can be set up for tight shots of my subject matter while the primary camera is fixed on me or a wide shot.
The real question in today's video world is not which camera, but HOW MANY do I use today?