I think the right map makes more sense. On the left, there is this outer rim of the map, that is actually very interesting to the setting and should not be split up, while on the right: Who needs details about the wastelands in the dayzone or the endless ice in the nightzone?
Two things that come into my mind: The day zone will be frozen over, if this has the base parameters like earth. Look at the north pole, it does no melt even in it's summer days, when the sun constantly shines on it at a low angle. When the sun does not shine at all, there should definitly be ice.
The dominant wind direction is easy: Air rises on the hot side and thus falls on the cold side and near the surface there will be a constant flow of air from night to day, thus the habitable rind will probably be a good bit towards the sunny side. Less "Land of the perpetual dawn" and more "Land of the perpetual 9 o'clock forenoon"