Sizing System for Screw Heads?

I am restoring an old lamp with a shade that attaches with sort of a hose clamp thing held on by a small machine screw. The screw was stuck so I ended up breaking the head off assuming it would be easy to replace.

Well I ran into a little problem. I have NO idea how you search for a specific size head. I went to my local hardware store. All their machine screws had heads almost twice as wide as mine despite being the right length and width. For a decorative item, those larger heads really wouldn’t look right.

Asking for a specific size head felt like I was speaking a foreign language. Can someone please inform me what terms to use when searching in person or online or what sizing system machine screw heads would go by? I thought this screw was fairly standard but I’m grasping at straws here.

The measurements are 5/8” long, 3/16” wide and the head is 7/32 or 1/4”.