Remove the door from the hinges, using a drill/driver. Place the door upside down across two sawhorses. Place a clamp at each end to secure it to the sawhorses.
Center the stained glass panel on the door. Trace around it with a pencil. Draw another tracing inside the first tracing, separated by 3/8 inch.
Drill four holes through the door, using the drill/driver and 1/2-inch drill bit. Drill the holes at each corner inside the smaller tracing. Alternatively, if the panel is oval or irregularly shaped, drill the holes evenly spaced around the perimeter.
Insert the blade of a jigsaw into one of the holes. Cut around the tracing from hole to hole, keeping the blade on the inside of the tracing. Remove the piece from the inside.
Install a 3/8-inch by 3/8-inch cutter bit -- with bearing at the bottom -- into a hand router. Place the router onto the door with the bit inside the cutout. Turn on the router. Pull the bit slowly into the wood until the bearing makes contact with the wood. Push or pull the router around the cutout in a clockwise direction, using both hands to cut a dado or channel around the inside perimeter of the tracing.
Sand the routed dado or channel smooth by hand, using 100-grit sandpaper. Dry-fit the stained glass panel into the routing. If it sticks or binds in the opening, continue sanding the routed opening until it fits. When proper fit is achieved, color the routed area, using a stain marker.
Apply a heavy bead of clear silicone caulking to the inside corner of the routing. Insert the stained glass panel into the routing. Place a book or moderately heavy object onto the glass panel to secure it and press it into the caulking. Alternatively, if the panel seems very fragile, use your fingers to press it into the caulking. Allow the caulking to dry overnight before removing the book or handling the door.
Hang the door back onto the hinges, using the original screws and drill/driver.