The trim piece is very much like a roof edge moulding. The top can be a piece of bullnose or a square 4 sides with a bevel cut down the edge that connects to the door. I have seen this on inside and out, usually with a moulding treatment at the top that somewhat matches. It's there to divert the water away from the glass and the wood panels. If you notice, the trim is mounted a little below the glass (or a little above,,), this is to prevent getting a nail in the glass. It's a cool old feature that is not so popular today in this fast paced "git'er'dun" society.
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