I am certainly no authority on the different styles is quilts/quilting. But like everyone else, I have my own opinions.
Traditional to me brings to mind things like double wedding ring, grandmothers flower garden, four patch, hand embroidered state flowers, and all the others from the era of our parents and grandparents, usually done in vintage type fabrics.
Modern brings to mind designs like rectangle in a frame, ones done primarily with jelly roll strips, Bargello, ones with lots of "white space" and blocks or circles of color.
Modern can also be traditional patterns done in more unique color combinations. Or would that be considered Contemporary?
I think of Art Quilts as being things like landscapes, portraits, thread painting, as well as wall hangings that are just "different".
Fiber art includes knitting, crocheting, embroidery of all types, weaving, tatting, lace making, braiding, etc. as well as quilting and any other type of sewing.
Can't answer anything about quilt shows as I've only been to two so far.
I do know that judging in the art category at a local county fair had been known to be very much determined by the judges taste. (For a number of years if it wasn't a floral landscape or a cowboy on a horse it didn't stand a chance. Lol!)