Err... no it isn't? The entire naming convention in wooden sailing ships of "Ship of the Line" was quite specificly about the ROLE of the ship: it fought in the main line of battle. It was then rated based on size. Likewise that's why ships that WEREN'T called "ships of the line" were called different thing. Frigates were named as such for their role as well, all "Frigates" were expected to fight mainly outside the line of battle, or act as independent raiders or patrol ships, rather than as part of a fleet. Even when a ship's class was based on its rigging et all, it still would end up with a partially role based name, IE "Sloop of War" were "sloops" (a classification based on rigging and hull design) designed and equipped for fighting (their role).
Yes, we got more detailed and specific with the classifications in the late 19th century, but role based names have always been part of ship classification.
@S'task that was a second parallel classification though and tended to be informal rather than the formal designation based on size. The only formal type designation was still regulatory, not role-based--"Post Ship", which meant it rated a Post Captain. Part of the reason for this is that what was a frigate and what was a ship of the line in the sense we think of did not match frigate and ship of the line in the terms of the age of fighting sail.
The Frigate was once most smaller ships of the line, with the larger ones being called "Great Shyppes" in the 17th to early 18th century--because line tactics themselves didn't exist. A relatively large group of two-deckers of 40 - 60 guns existed after this point which were built exactly like small SOLs but deemed too small to stand in the line, and were used in frigate-like secondary roles; these ships were unambiguously larger than the classic frigates of modern conception of the same era, however. They also lasted straight through until the end of the 18th century.
To elaborate on
@Captain-General's response, I'd say it really depends on the timeframe, the theater, and often the individual mission. And technology, of course; time period is critical for just what that rate or guns mean.
Ships were generally designated based on their rig and armament, though most non-ship rigged ships were below the rate. I've seen lots of period and semi-period literature refer to a ship by her name and her (nominal) guns, since some guns counted and some didn't for rate. Bombs and fireships were really the only ships designated based on role, and even that wasn't hard and fast. The British system of rates is the most famous, and was constantly updated to reflect the increase in both number of guns and size of guns. For a while it was nominally divided between the first four rates, and the last two plus the unrated ships. But you saw the fourth rates go out of fashion in the 1700s, and then they came back as super frigates of 56-60 guns in the 1810s.
The roles weren't hard and fast, except as dictated by size. You wouldn't throw a brig into the line of battle against a third rate, but nothing prevented said third rate from running down that brig if fortune favored them. Using your sloop example, they were put to a multitude of tasks, which would also be assigned to cutters, brigs, schooners, or frigates, or even to ships of the line if necessary. You would see a fourth rate in the line of battle at Minden in 1756, but also serve as Anson's flagship on his round the world cruise in 1740. Then they were back in the line of battle, probably out of necessity, at Navarino.
While British third-rates as commerce-destroyers were not that common, you did see them in what we might consider "cruiser work," as convoy escorts, part of the inshore squadron, etc. A good example of a ship-of-the-line doing "frigate work" was POITIERS, who took WASP and recaptured FROLIC after their duel when FROLIC was escorting a convoy. The British might also have a "flying squadron" much like the fast battleship or battlecruiser squadron of WWI, charged with running down or engaging damaged or the rear of the enemy line. The British would also send out ships of the line much like they did Sturdee in 1914, to hunt down a powerful enemy raider or small squadron. LONDON (a 98!) took MARENGO in that kind of operation.
Ships of the line as raiders were more popular as raiders in France and the United States. MARENGO under Linois is one good example. Another might have been the US ships of the line. Lacking a battle fleet, there wasn't much we were going to do with AMERICA but raid. The 74s we built right at the end of 1812 and then under the Gradual Increase usually served as foreign flagships and leaders of frigate squadrons, but the OHIOs were also fast as thieves, so heavy commerce raiders was entirely possible. They were sweet ships, so sad so many of them had such short lives.
Conversely, you saw frigates behave like ships of the line when the situation called for it. All of the lakes battles in the War of 1812 worked that way, as did the Adriatic operations under Hoste. The entire Mauritius campaign saw frigates (mostly fifth rates) operate in all the roles, from scouts, to convoy escorts, to a battle fleet. In each situation, the role was more determined by availability than size. If anything, Rowley considered RAISONNABLE (64) to be a liability, not an asset, off Mauritius.
Rate also wasn't necessarily a guide to the seniority of her captain, either. Flag captains were often the most junior captains in the fleet, and captains of first, second and third rates were often junior to captains of fifth and sixth rates. Because command of a "cruise" was a very profitable endeavor, but also put a captain outside the admiral's control and thus put a premium on his judgment.
So I think rate as a ship's role was more a guideline than a rule, except that you're not likely to see single mast cutters taking on first rates in pitched battle. It guaranteed a rough idea of guns and manpower vs. a contemporary, but could easily be assigned the same tasks as a ship much larger, or smaller, as necessary.
