The problem is that there's zero information about what qualities a starship has that makes it be classified as a light cruiser, heavy cruiser, medium cruiser, frigate, (insert naval term here), etc. The Constitution class, the Ambassador class, and the Curiosity class were all referred to as heavy cruisers, but there's no indication as to why this is so, or why, say, the Galaxy class is not classified as a heavy cruiser.
The terms ought to be comparable to current naval terms. I mean, a heavy cruiser is a heavy cruiser and a frigate is a frigate.
https://en.wikipedia.org/wiki/Light_cruiser
A light cruiser is a type of small- or medium-sized warship. The term is a shortening of the phrase "light armored cruiser", describing a small ship that carried armor in the same way as an armored cruiser: a protective belt and deck. Prior to this smaller cruisers had been of the protected cruiser model, possessing armored decks only. While lighter and smaller than other contemporary ships they were still true cruisers, retaining the extended radius of action and self-sufficiency to act independently across the world. Through their history they served in a variety of roles, primarily as convoy escorts and destroyer command ships, but also as scouts and fleet support vessels for battle fleets.
https://en.wikipedia.org/wiki/Heavy_cruiser
The heavy cruiser was a type of cruiser, a naval warship designed for long range and high speed, armed generally with naval guns of roughly 203 mm (8 inches) in caliber, whose design parameters were dictated by the Washington Naval Treaty of 1922 and the London Naval Treaty of 1930. The heavy cruiser is part of a lineage of ship design from 1915 through the early 1950s, although the term "heavy cruiser" only came into formal use in 1930. The heavy cruiser's immediate precursors were the light cruiser designs of the 1900s and 1910s, rather than the armoured cruisers of the years before 1905. When the armoured cruiser was supplanted by the battlecruiser, an intermediate ship type between this and the light cruiser was found to be needed—one larger and more powerful than the light cruisers of a potential enemy but not as large and expensive as the battlecruiser so as to be built in sufficient numbers to protect merchant ships and serve in a number of combat theaters.
So, heavy cruiser and light cruisers are both cruisers. The heavy and light refers to the amount of armor vs. speed.
Frigate vs. Destoryer
“Frigates are thus usually used as escort vessels to protect sea lines of communication or as an auxiliary component of a strike group whereas destroyers are generally integrated into carrier battle groups as the air defence component or utilised to provide territorial air and missile defence.”
https://www.naval-technology.com/features/frigate-vs-destroyer-difference/
Realistically in today's navy a destroyer would fill the roles we see the Enterprise (especially in TOS) perform.