But if you insist on the technical, a cavesson is not just the noseband, the term includes the strap that goes over the head, the headstall. So a cavesson is a headstall, separate from the bridle, with a noseband attached.
But no, the cavesson does not only come in a plain, flat style and in general the terms are used to mean the same. There are several types of cavessons, as I listed above. Check back with Dover's you will see them used randomly.
But - since you ask, I am from America! What you show as a "grackle" is what we here call a figure 8.
As you asked - here is a "definition" for my info:
"A noseband is the part of a horse's bridle that encircles the nose and jaw of the horse. InEnglish riding, where the noseband is separately attached to its own headstall (headpiece), it is often called a Cavesson. In other styles of riding, a simple noseband is sometimes attached directly to the same headstall as the bit."
Now the link for a "sourse" of the definition of the cavesson and it's use & some history:
Define Cavesson | Dictionary and Thesaurus
Please, I find it strange that you keep on with this subject, but I hope you are bringing this up because you are that interested.
I also hope that you continue to learn about all the different type of equipment and their uses. Knowledge is always good