I would say go with a Lichess study..because you can annotate the games and analyze and everything gets saved, so that the student can go through at leisure also. All you need to get right is the voice with a program of your choice.
Video is quite unnecessary, I would say.
@lovlas curious- what's so bad about teamviewer? Are there good biz alternatives that you know of?
Discord/Skype + Chessbase/Lichess studies works perfectly for me - I very rarely use any video, I don't think it is necessary to "emulate" a live session in this sense - in many ways doing an online session is more efficient, in my opinion. I do live sessions if I want to get to know the student a bit better - but if it's a student I am familiar with (and most importantly; the student is familiar and comfortable with me as a coach), then online sessions are my preferance, actually.
Sorry for the long answer :)
Generally if a product is free, you are the product. I wouldn't trust a for profit like teamviewer with handling your private data. Also its obsolete when lichess studies already does what it does and more. You don't need to share your entire screen, just the chess board. Studies can handle that.
@Pawnstructures "My question is if anyone here know a more practical way of splitting the screen view between Skype video and the chessboard? Maybe there is some software that can do this to give a more professional look than doing it by hand?"
There are third party apps that split up a desktop so that active windows 'snap' to a selected size and orientation. But it only affects your own view. I don't think a student will see the difference unless you somehow share the view of your desktop.
Well that should work because i can share view of my screen via Skype, also have OBS Studio maybe it can do something similar.
Regards Richard