DISCUSSIONS on “Rules for online games in HoMM3 and formulas for calculating player rating.”

"each player has the right to take a restart/restarts during their turn"
Is it not allowed during the opponent's turn?

Well, it means that you can take 111 restarts until you pass your first turn. 113 - until you pass your turn on 113, you can do it earlier, but not later.
And so - you can take 113 restarts during the opponent's turn on 111.