Think of the tray as a drawer bottom. The slot will be as wide as the tray is thick. Grain is left to right. The tray can be made two ways. One is the same thickness from end to end, fitting into the slot. Or you can have tenons with very narrow shoulders.
Thinking of the tenon approach you can cut back the tenons at the front side of the tray. This will create the illusion of a perfect fit. The picture below shows some trays I put in a credenza. These trays are the same thickness as the slot with the front edges cut back about a half inch. The slots are about 13" long while the tray is 13.5" front to back. In your project just cut the rear groove deep enough to allow for some wood movement
