Here's the difference on the '63-6 vert W/S gasket

Who knows what could have happened to your car over the last 55 years. Originally they had a soft gooey sealer on the gasket to frame sealing area. The gasket to glass didn't have anything near as I can tell. Sometimes glass installers will do what they think is best, not necessarily what the factory did. On installation, I've used both the gasket first and then the glass method, and also the glass and gasket together, and install with the rope method. I believe the former is the way the factory did it. It's less messy inside and you can better seal the gasket to the frame that way.