Lost in translation, in the uk, cap screw usually refers to socket head cap screws only. Hexagon head screws and bolts are usually called hex head.
From Machinery's Handbook, 25th ed., pp. 1417: "A bolt is an externally threaded fastener designed for insertion though holes on assembled and is normally intended to be tightened or released by torquing a nut. ..... A screw is an externally threaded fastener capable of being inserted into holes in assembled parts, of mating of mating with a preformed internal thread or forming its own thread and capable of being tightened or released by torquing the head."Cap screws usually have threads the entire length of the bolt. Bolts have threads on roughly 1/3 - 2/3 the bolt length
From Machinery's Handbook, 25th ed., pp. 1417: "A bolt is an externally threaded fastener designed for insertion though holes on assembled and is normally intended to be tightened or released by torquing a nut. ..... A screw is an externally threaded fastener capable of being inserted into holes in assembled parts, of mating of mating with a preformed internal thread or forming its own thread and capable of being tightened or released by torquing the head."
Both Fastenal and Grainger sell both hex head screws and hex head bolts in full or partial thread configurations. For some reason, McMaster Carr doesn't sell hex head bolts, only hex head cap screws in either full or partially threaded configuration They do sell carriage bolt which are fully threaded for the most part. Go figure.
I think your last option is most likely. The original meaning of "bolt" was a short arrow as in a crossbow bolt. It came to mean a short metal pin used to fasten objects together in the 1400's. Adding a head and threads was an obvious improvement.If you torque it from the head its a screw
If you torque it from a nut it is a bolt
So, what is the design difference between a hex-head bolt and a hex head screw?
Is there some fillet that is on the screw and not the bolt? or vice versa ?
Is there some relief that is on the screw and not the bolt? or vice versa ?
Do you use a different washer on the screw or bolt.
Or is this something that has passed into history?
The most common reference is that a screw is threaded to the head and a bolt has a plain portion, any other references have no practical use.If you torque it from the head its a screw
If you torque it from a nut it is a bolt
So, what is the design difference between a hex-head bolt and a hex head screw?
Is there some fillet that is on the screw and not the bolt? or vice versa ?
Is there some relief that is on the screw and not the bolt? or vice versa ?
Do you use a different washer on the screw or bolt.
Or is this something that has passed into history?