There seems to be a very blurry distinction made between destroyers, frigates, corvettes and even cruisers these days.
The Hobart is based on a Spanish frigate. The Japanese ASW helicopter carriers are called destroyers and as you said the US redesignated its Cruisers to frigates and then back again. Damens Sigma class can be either frigates or corvettes.
Australia seems to define destroyers as AAW vessels and frigates as ASW or GP ships.
It is all very confusing.
If you look back in history the first "cruisers" were actually frigates, corvettes and sloops, i.e. non ships of the line capable of independent cruising, trade protection, and gunboat diplomacy. Frigates were the largest and fastest, HMS Warrior the RNs first iron hulled battleship and the most powerful warship in the world when completed, was technically a frigate, or a large fast cruiser.
As technology evolved allowing sails to be deleted cruisers evolved into distinct battleships, armoured cruisers and protected cruisers of varying sizes. Torpedo boats also evolved at this time and their counter, the torpedo boat destroyer, was also developed. Torpedo boat destroyers became simply destroyers, cruisers evolved into Battlecruisers (effectively replacing armoured cruisers) an light cruisers of various sizes, ranging from the long range trade protection type such as the various Town classes (HMAS Adelaide I was the last of this type completed) through smaller fleet and scout cruisers, some were very much destroyer killers and destroyer flotilla leaders and others basically super destroyers.
The Washington Treaty of 1921 introduced the Heavy Cruiser with a maximum or 10000ton standard displacement (calculation of which was defined by the treaty) and a maximum gun calibre of 8". The prototype was basically the RN Frobisher Class, an enlarged evolution of the Town Class, by way of the "Atlantic Cruiser" design studies for a new trade protection as opposed to fleet cruiser type. Prior to the treaty the RNs vision for a suitable trade protection cruiser (and thus suitable for dominion navies) was 20000tons, 9.2" (if not 15") guns, torpedo boats on davits and hangar + catapults for scouting and torpedo attack float planes.
The 1930 London Naval Treaty introduced a cap on total cruiser tonnage for each signatory nation and a new maximum calibre of 6", mostly as a cost saving exercise. This is where the 8000 ton 6" gunned light cruiser came from, the type we most think of when someone says cruiser.
During the war cruisers again grew with the last US types displacing 20000tons, as much as the first Dreadnought Battleships. Frigates re-emerged as improved convoy escorts to supplement and replace the smaller, less capable corvettes. Sloops had been introduced in the treaty environment as slower general purpose ships not restricted as destroyers were and evolved into highly capable ASW and mine counter measure vessels. Interestingly the US called their frigates Destroyer Escorts (DE) a terminology Australia adopted for our River Class (modified Type 12) frigates.
Post war the RN retained the status quo in classifying frigates as specialist escorts good at one thing and mediocre at others, destroyers were general purpose ships that were good at everything, sloops were general purpose ships that were slower than destroyers and mediocre / good enough at everything. Cruisers had command facilities and could support themselves away from base, i.e. didn't require the support of a destroyer tender.
The US took another tack, continuing to use the DE terminology for what everyone else called frigates and reintroducing the term Frigate for their new generation of large destroyers or destroyer leaders. These frigates (DL for destroyer leader) evolved into guided missile frigates (DLG), with the large RN County Class guided missile destroyers often also being referred to as DLGs. In 1975 the USN brought their designation system inline with NATO reclassifying their larger, newer DLG and DLGN (nuclear powered) frigates as cruisers, the older smaller ones as DDGs and their various DE and DEG as FF and FFG.
Ironically current generations of frigates and destroyers have more in common with the cruisers of the past, in size, capability and role. So what is in a name?
