how do you glue a piece of decorative wood to a painted door?

It depends how heavy this piece you want to glue is. Paint doesn't hold all glues that well, as you noticed.
To get a better grip, you may want to sand down the areas on the door that will have the glue adhere to.
I would use wood glue, good ol' Elmer's.
I would take the door OFF the hinges though, and lay it flat somewhere. THEN glue the piece on so it sits on top of the door overnight. Then rehang the door.
 
You don't have to use glue-- if the decor piece is light as stated than go to the store, hardware or craft and buy some super strong double sticky back tape-- this will work--
before you apply it make sure you wash the area well , then dry--- you will not be disappointed
 
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