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

but that's not so obvious... because I believe that if an opponent buys him out on that day, then he has the right to perform 2 Dimension Doors. Or do you think this is a technical defeat for the opponent?

But the opponent hasn't performed 2 Dimension Doors with this hero yet. So everything is obvious.

Let me explain in more detail:
"the rules forbid each hero from performing more than 2 Dimension Doors a day" - incorrect statement
"the rules forbid a player from using more than 2 Dimension Doors per day with each hero" - correct statement.
Prohibitions and restrictions in the rules are imposed on players; the player bears the responsibility, not the hero.