Ship classification terminologies have always been in flux thanks to the constantly-evolving nature of naval warfare. The word “frigate” originally carried a connotation of speed and applied to ships not intended for the battle line; later, a frigate was any warship which carried its guns on a single deck, so many early ironclads were technically frigates. In modern times, surface combatant classification has centered around displacement instead of characteristics such as sail configuration or number of gun decks. In this size-based hierarchy, a frigate is a warship which is smaller than a destroyer but larger than a corvette.
During the Cold War, most frigates displaced around 2,000 to 4,000 tons at full load and were designed for escort duties. They tended to have a light armament, with point defense anti-air missiles, a small naval gun, torpedoes or depth charges, and perhaps a few surface-to-surface missiles, depending on the class. The American Knox-class frigate, for example, had a trainable launcher for eight Harpoon anti-ship missiles and/or Anti-Submarine Rockets (ASROCs) but relied on the Basic Point Defense Missile System (BPDMS), a lackluster setup which consisted of Sea Sparrow missiles and a manually aimed radar illuminator, for air defense. The British Type 21 frigate used a similar manually-guided anti-air missile system, the infamous Seacat. The Type 21 was not originally designed with any other missile systems, although it was quickly fitted with four Exocet anti-ship missiles following criticism of the design’s light armament. There were a few frigates with medium-range surface-to-air defense missiles, such as the Oliver Hazard Perry class, which was equipped with a Mk 13 rail launcher and the Standard SM-1. But, by and large, most frigates were smaller vessels lacking medium and long-range anti-air missile systems, and even classes such as the Oliver Hazard Perry had equipment substantially inferior to a contemporary guided missile destroyer.
However, as the Cold War drew to a close, surface combatant technology made considerable progress. Long-range radars became smaller and cheaper, as did the electronics needed to operate them, permitting smaller vessels to perform the air defense role. Vertical launch systems (VLS), which store missiles below decks in a ready-to-fire state instead of launching them from a rail or an above-deck box, rapidly replaced rail launchers. Vertical launch systems provide higher-density weapon storage, allowing smaller ships to carry medium and long-range air defense missiles.
Thanks to such advances, high-end missiles and electronics could be fitted to smaller ships, spawning a new breed of multirole surface combatant. Such vessels, while classified by the navies which operate them as “frigates,” are nearly as large as a small destroyer and have remarkably similar capabilities, including the ability to perform area air defense. Examples of such frigates include the German Sachsen class, the Franco-Italian FREMM, the Spanish Álvaro de Bazán class and related Norweigan Fridtjof Nansen class, the Danish Iver Huitfeldt class, the Dutch De Zeven Provinciën class, the Russian Gorshkov class, and others. Many of these ships are the most powerful surface combatants in their respective navies. As this list suggests, the practice of commissioning frigates with destroyer-like capabilities is largely a continental European phenomenon — many of the aforementioned ships would be classified as destroyers if operated by a non-European navy.

An Álvaro de Bazán-class frigate. Image by Hammersfan.
Take, for example, the Spanish Álvaro de Bazán-class frigate. The class boasts a full-size AN/SPY-1D air search radar, the exact same radar used aboard American Arleigh Burke-class destroyers. Its primary armament is a 48-cell Mk 41 VLS, also used aboard Arleigh Burke-class destroyers, which is compatible with a wide variety of American air defense, surface-to-surface, and anti-submarine missiles. The biggest difference between the Bazán class and the Burke class is the latter’s greater number of VLS cells — 90-96 vs. 48. If you compare the Álvaro de Bazán class to a smaller destroyer such as the British Type 45, the firepower discrepancy vanishes, with both ships having 48 VLS cells.
However, smaller, more specialized frigates have not yet died out. South Korea’s Incheon-class frigate, a ~3,000-ton coastal defense class designed for the unique needs of the ROK Navy, illustrates this. It has a large Mk 45 Mod 4 naval gun and four Cheon Ryong land-attack cruise missiles, allowing it to conduct naval gunfire bombardments along the coast and strike shore targets in North Korea with precision. The class’s eight SSM-700K missiles offer potent anti-ship capabilities, and its facilities for a medium helicopter grant formidable anti-submarine prowess for a ship of its size. However, its anti-air armament is a single 21-round Rolling Airframe Missile (RAM) launcher, which has a paltry range of less than 5 nmi. Compared to the aforementioned multirole frigates, the Incheon class is smaller, cheaper, and optimized for a specific role — land attack.
One of the reasons ships are given descriptors such as “destroyer” and “frigate” is to serve as a heuristic for quick analysis. When told that a ship is a destroyer, one has a rough idea of its capabilities without needing to dig into the specifics. However, it is not possible to make such assumptions with regards to frigates — there are such a diversity of frigate displacements and capabilities that closer examination is essential to get a sense of the role and capabilities of each individual class.
This is far from an ideal state of affairs, but there is no real alternative. Ships are classified by the navies which commission them, and, for the sake of consistency and clarity, everyone uses the given classification. As such, naval enthusiasts and analysts should recognize the diversity of frigates and be vigilant when considering frigate types with which they may not be familiar.
As you said, classifications and nominations of different ship types have changed over time, but the “smaller than a destroyer, but bigger than a corvette”-definition for a frigate always struck me as especially arbitrary and nonsensical.
After all, destroyers started out as “motor torpedo boat destroyers”, a class of ship specifically designed to defend bigger ships from attacks by small speedy motor torpedo boats and the first few generations of destroyers actually weren’t much bigger than the boats they were meant to hunt down and were often even smaller than corvettes.
Over time, destroyers became bigger, obviously, mostly to enable them to escort convoys over long distances, but where this modern common knowledge that they need to be bigger than frigates and corvettes to qualify as a destroyer comes from, baffles me. It seems to me that both in classification and in supposed use, modern destroyers are actually light cruisers.
As you touched on in the first paragraph of the article, the definition of a frigate originally had nothing to do with size. Frigates were basically just battle ships that were not supposed to fight at the front of big battle formations, but were supposed to operate alone or in small groups and do long range patrols and thus favored speed and endurance over toughness and fire power. One could argue that according to the old, original definition, most of the big German WW2 ships, like the Sharnhorst and Gneisenau, or the Deutschland class and in a way even the Bismarck class, were actually frigates.
Long story short, when you hear people say “This frigate is so big, it almost is a destroyer”, that is really weird to someone who has the original definitions for both classes in mind.
Comparing the Bazan Frigate to the Type 45 Destroyer ignores the fact that the Type 45 is significantly heavier, is 5 knots faster, and has a range of 2k nautical miles more. And then there is the sad loss of the Norwegian Frigate Helge Inqstad which suggests that maybe the design is not as strong as it should be. That later loss is likely to be one of the reasons that the Bazan design was not selected as the basis for the US Navy’s new FFG(X).