The first ships commonly thought of as 'fast battleships' were the British Queen Elizabeth class, constructed from 1912-1916. Unlike later fast battleships, these had a distinct tactical role envisaged for them. British tactical though in the pre-WW1 period emphasised the need for a 'fast wing' of the battlefleet. This would be a fast squadron that could threaten the front and rear of an enemy's line during a fleet action; either the fast wing would achieve a superior position relative to the enemy's line, or force them to manoeuvre such that they were an inferior position relative to the main British line. This role had originally been envisaged for the pre-dreadnought armoured cruisers. Once the dreadnought age rolled around, battlecruisers naturally took this role (see here more information on what a battlecruiser is). However, as the Germans began to build their own equivalents, exercises showed that the British battlecruisers would be too busy engaging them to form a fast wing. Admiral Callaghan of the Home Fleet described the results of these exercises:

experience gained… appears to show that the battle-cruisers will be occupied in advance of the battlefleet… if so, it is not improbable that they will also be engaged with the enemy’s battle-cruisers and cruisers, and may consequently be unable to form on their own battlefleet before the general action commences.

This pointed to the need for the fleet to have its own 'organic' fast wing as an inherent part of it. The Queen Elizabeths were constructed to fulfil this role. They had a top speed of 25 knots, four knots faster than the typical British fleet speed of 21 knots, but three knots slower than the contemporary battlecruiser Tiger. They also had the same armour and armament as contemporary battleship classes, like the 'R's. As such, considering them 'fast battleships' as a distinct class makes perfect sense - they were an intermediary between the battleship and battlecruiser, both in terms of speed and role. However, they were also as fast as older battlecruisers, leading to some to argue that they should be considered as such. Admiral Jellicoe, seeking to strengthen the British battlecruiser force at the outbreak of WWI, would make this argument in a letter to the Admiralty:

In order to give a proper superiority in fast armoured ships, the four Queen Elizabeths should, I think, be classified as fast battlecruisers, bringing the respective totals of these fast ships to - Germany 8, Britain 13.

At Jutland, the Queen Elizabeths would fight with the Battlecruiser Fleet. However, this was because they had been seconded to this force to replace the 3rd Battlecruiser Squadron (which had been sent to Scapa Flow for gunnery practice), rather than because Jellicoe's earlier arguments had borne fruit.

The next British ship that might be considered a 'fast battleship' is the Hood. The first designs for Hood (and her three cancelled sister ships) were along the lines of the classic battlecruiser. It would have 32 knots of speed, but only had an 8in belt (though this was fairly extensive, had an effective thickness increased by its slope, and was backed by a 1.5in shell plate). Her keel was initially laid down on the 31st May 1916. Other events occurring on the same day led to a major redesign. The explosions of three British battlecruisers at the Battle of Jutland were seen as discrediting lightly armoured battlecruisers. Instead, the Admiralty wanted thicker armour, at the expense of speed. Hood was redesigned with thicker armour protection than the Queen Elizabeths (the belt thickness was the same, but Hoods was sloped and again, backed by a 1.5in plate). Hood is often described as a battlecruiser, because that is how the RN classified her, and the tactical role she was used in. However, if we look at just her physical characteristics, the argument that she is a fast battleship becomes more clear. Some 33% of her displacement was armour, a greater percentage than any British battleship bar the King George V class of 1936. Her armour protection was equivalent to contemporary British battleship designs.

The immediate post-WWI period saw a profusion of battlecruiser designs that might also be considered fast battleships. The British G3 design is particularly notable, having the same armour and armament as the American Iowa class of twenty years later. However, the Washington Naval Treaty of 1922-3 brought an end to these designs. The building holiday enforced by Washington would be extended until 1936 by the First London Treaty of 1930. When it expired, most major navies constructed new battleships. All of these ships had speeds in the vicinity of 30 knots. They were also all indisputably battleships, built as the centrepiece of the battlefleet, rather than as a peripheral but significant element like battlecruisers. All the classes had thick armour, and heavy armament, making them physically battleships too. With their high speeds, they were much faster than the older battleships that made up the majority of the world's navies. As such, they are described as fast battleships. The Japanese Kongo class, originally built as battlecruisers in 1911-15, would receive heavy rebuilds in the 1920s and 1930s. These rebuilds greatly increased their armour protection, and combined with new boilers, would put them onto the same standard as these newly built ships. Hood had a similar rebuild planned, but would be sunk before it could be carried out. These rebuilt ships are similarly classified as fast battleships, as they have the same capabilities and were used in the same ways as the newly built ships.

Pocket battleships, meanwhile, are usually limited to just a single class of ships: the German Deutschland class, built in 1929-36. These were ships that didn't fit into the usual classification schemes used at the time. They had six 11in guns, an armament in between the 14-16in guns of contemporary battleships and the 6-8in guns used by contemporary cruisers. They had a speed of 26 knots, significantly quicker than most battleships in existence at the time, but slower than cruisers and battlecruisers. The Germans called them 'panzerschiffe' (literally 'armoured ships'). This classification wasn't used by the British, who coined the term 'pocket battleship' to describe them. The Deutschlands were constructed to be merchant raiders. Their armament and protection was sufficient to fight off any cruiser, which were the ships typically used to protect trade. They were fast enough to outrun most battleships, which outclassed them in firepower. While battlecruisers might be able to defeat them, the Deutschlands were mainly aimed at the French, who had none. This role was a classic cruiser role, and so I tend to think of the Deutschlands less as 'pocket battleships', and more as 'large cruisers'. This is a distinct classification that can be used for several classes, either constructed or designed, in the run-up to and during WWII. They are ships that were built to carry out the cruiser role - merchant raiding/protection and scouting - but were better armed and armoured than a heavy cruiser (which by definition has 8in guns). They were not intended to form part of the battlefleet, and were not expected to be capable of fighting battleships as a matter of course, meaning they are neither battleships nor battlecruisers. The later French Dunkerques (a response to the Deutschlands), German Scharnhorsts and American Alaskas are the other classes I consider 'large cruisers', as they all have similar characteristics and doctrinal usage to the Deutschlands.