I have a double kitchen sink now and am replacing it with a large single sink. The current sink has a disposal in one basin and a standard drain in the other. The disposal is connected to a trap and then drains through the floor. The other basin is connected to a trap and then drains through the back wall of the cabinet.

Is there a preferred drain to use? Either one looks like it could be plummed without issue. The other one will be trimmed and have a cap cemented in place.

For background: The sink is under a window and the space below it is finished. I believe that the installation is original to the 1981 house. All of the drains in the house are PVC and there isn't a vent in the roof over this (or any other) exterior wall, the closest vent is above the bathroom approx 15' from the sink.

use the one in the wall.

IF the disposer really drains into the floor, it is an improper and illegal connection. Either use the one in the wall, if it is low enough, or revise the other one to make it a proper connection.