If you do a Google image search for shepherd’s crook you will find quite a variation of shape in what people consider a crook.
For some reason I always thought of a hook as being shaped like most fish hooks - a simple J hook with the shank leading into the end hook on the tangent line. To me the name changed to a crook when there was a kink bend at the tangent point to bring the shank into a line pointing to the center of the hook arc.
Usually when I need a short hook I make an S hook. If I need a long hook I make a crook type. These extended hooks are almost always used outdoors to hang nest boxes, feeders and garden ornaments. The largest use 1/2” round stock and have approximately 5” diameter crooks and are 2’ to 4’ in length. The throat opens enough to pass over the limb being used. I use them to hang large nest boxes. Tis has been a great predator deterrent.
My smaller hooks are made from 5/16” round. I make them in various lengths to hang feeders near the house. Having a collection of various lengths makes it easy to adjust the height for easy filling and the best viewing. The end crooks on these are approximately 3/4” diameter. The throat opening just 3/8” or so.
Anatomical terms can be confusing when describing the crook. Is the gap in the ring the mouth or throat? Is the open center of the hook the eye? It may be best to describe the opening to the hook in degrees from the center of the radius - open 60º or open 100º etc. Or we could use the clock-face analogy. Most of the hooks I make have an opening between 2 and 10 o’clock or between 1 and 11 o’clock. Just measuring the opening may give the clearest description.
I went into quite a bit of detail about making S hooks but I don’t have much to add on crook hooks.
One difference is in how I deal with the tip. The S hook tip is just a plain round taper. In a crook hook I start the taper where the arc begins and usually often end with a tiny lip curl.
Probably more than you’ll ever need to know about sticks.