Need the name of a small pin used as a cheap door hinge

No answer, but some ideas. Make sure the hinge is properly installed. It sounds like it was installed in the wrong orientation or the wrong type of hinge. Some hinges are design to be installed on lift up type cabinet doors. The spring action helps make the door feel less heavy and not slam shut. If you have other ones that work properly, check it against the others. The picture you have shows only one screw which leads me to believe that it was designed for another type of door. Overtime, I can see the door coming out of alignment by pivoting on that single screw.

If you are certain it is properly installed, then find the spring mechanism and try to play with it using needle nose pliers or a screw driver. If no luck, then try to disable or remove the spring.

Resetting the hinges so the door sits a little further from the stop will work. Sometimes it is easier to remove the door stop and reset it away from the the door a bit, especially if you are refinishing the door jamb anyway