Replacing Glass on a Door with Wood Mouldings 

It's actually a very common moulding found at most lumber supply and or home improvement centers. Commonly referred to as (13/32") panel moulding (or chair moulding, depending on ones geography)

Good question - there's usually only one side of the door that has removable moulding, the other side moulding is actually part of the door. Check for small gaps, nail holes or their impressions, indicating attached moulding. Also, you can test for gaps, and penetrate sealants with a razor blade to make sure you're working on the correct side.

Did you ever find a solution? My door is the same.

@@TheSwankynina Hi, yes, I used a really, really small screwdriver to lever out the pins and replaced them with brass screws which don't look too bad as I have brass door handles. I'll probably replace the door at some point but it's OK for now. Nobody notices! 😁