Am I correct in understanding that in order for the game to use arenas created by me for regular (non-scripted) battles on my map, I will have to create a separate mod?
Most likely, yes. There is a mod for new arenas for dueling battles; I recommend asking its author about the creation process.
Added 2 minutes later
Please excuse me, this panel is not displayed for me, and I don't know how to open it. Could it be that I have a different version of the editor?
No, you simply haven't enabled the advanced mode of the map editor. You can find out how to do this either in my Manuals (in my signature) or here => /topic/at/483279/
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