To be sure, even today these designations are more or less meaningless and interchangeable. A given ship may equally well be classified as a frigate, destroyer or cruiser, because all of these perform the exact same mission: do everything at once, sometimes solo, sometimes as part of a fleet, in order to maintain sea supremacy.
Smaller navies just can't afford seagoing ships that wouldn't be doing everything, from submarine or aircraft defense to sea and land attack. Larger navies used to be better off, but no longer are.
It's the British who started this whole mess. Back before WWI, things were simple: there were armored capital ships or battleships that would pound each other in fleet action; poorly armored secondary ships that would cruise solo or in small groups, hunting for soft targets; and destroyers that would both use the new weapon called self-propelled torpedo to threaten the capital ships, and defend the capital ships against such use.
However, the world wars called for quickly constructed escort vessels for antisubmarine and later anti-aircraft work, and there was a whole range of sizes and capabilities for such vessels, depending on where and how they were expected to operate and how much the navies could afford. For the multitude of designs, the Royal Navy resurrected obscure and illogical names that had formerly applied to sailing ships: frigate, sloop and brig returned to naval terminology, in no way logically matching their earlier usage.
It has been a mess ever since. Today, there are just three types of surface warship: the aircraft carrier, the amphibious attack ship, and the fighting vessel. The first two are partially merged in some navies. The last can be alternately called frigate, destroyer, cruiser or sometimes corvette; the first three are utterly uninformative, while the last vaguely suggests a somewhat smaller size. The Soviets understood the futility of this, and abandoned the old names, save for adopting "cruiser" as the synonym of "very large" in a system where designations ran from "boat" through "small" and "large" to this "cruiser/very large". Other navies tried to do a similar trick and define frigate, destroyer and cruiser as size indicators, usually in this ascending order. However, few navies could afford two different size categories of vessel, and they would arbitrarily pick one of the names for their modern capital ship. Many NATO navies had frigates only, although the Royal Navy insisted on calling some ships destroyers. The USN first had frigate as larger than destroyer, then switched this when more cruisers were needed for PR reasons and every frigate was redefined as a cruiser, then small "destroyer escorts" smaller than the average destroyer were redefined as frigates.
Would Starfleet try to maintain the futile idea that the names indicate size? Dubious, since nobody has ever made a working system that would logically associate the names with sizes. Would Starfleet use historical names relating to the missions of these ships? Dubious, since frigate and cruiser are the same thing historically, yet Starfleet uses both - and the very specific missions of some of the historical types have disappeared when naval technology has changed, and are unlikely to return.
I guess the best bet would be to assume that Starfleet has invented whole new definitions for these historical names, definitions that make sense for an interstellar navy that has major nonmilitary commitments.
After that, those naval terms were never used again to describe Starfleet vessels
Well, "destroyer" made its first appearance in late DS9. The idea of separating the present from the future wasn't quite completely upheld...
Federation doesn't built ships of the line, just different-class frigates.
Which is apt for two historical reasons: it echoes the doctrinal choices of the real United States Navy that was close to the hearts of the people writing Trek, and it echoes the type of old naval adventure that most often was glorified in literature. Ships of the line seldom featured in adventure books, except in the hands of villains and buffoons: the heroes of Patrick O'Brian or C.S. Forester commanded frigates of various kinds.
Timo Saloniemi