on the first day of my music history class our professor defined "Classical" as being anything from the mid 1700s-mid 1800s and "classical" (referred to as 'little c classical' during lectures where it is difficult to pronounce capitalization) as the colloquial usage that basically means anything between Gregorian chants and the Harlem Renaissance

This is what I was taught:

There are two types of music: classical and folk.

Classical music involves musicians having a high technical proficiency with their instrument and the ability to translate musical notation on paper into music through their instrument. Such musicians are said to be classically trained.

Folk music also involves musicians having a high technical proficiency with their instrument; however, they rarely translate musical notation on paper into music through their instrument. Instead, they tend to memorize musical structures and are skilled at composing on the fly, i.e., improvisation.

Folk music predates classical music by thousands of years. Both types of music have several genres. Classical has baroque, classical, romantic, modern, and contemporary. Folk has pretty much everything else! In this context, folk music is much more than American folk, bluegrass, and country: it also encompasses rock, blues, jazz, and pop, for example.