I'd like to create tournament for my students where I specify the starting positions.
Which websites allows users to create tournaments with different starting positions?
You can do this
at https://lichess.org. Go to the "Play" tab, and select "Tournaments." Assuming that you have a lichess account, (all they need for that is your e-mail, and they do not spam it) you will see an option to create a tournament. Once you select that option, you will be able to specify the start position in the tournament (this position must be an opening with an ECO code).
If you wish to have only your students playing in the tournament, you can make the tournament private (password required to enter) and send the password to your students.
You can also play a game (not tournament) from any legal position (opening, endgame, middlegame, whatever) via the board editor, which is under the "tools" tab