The problem is that there is no standard classification for battlecruiser - they're not like heavy cruisers, which were defined by various treaties as ships with 8in guns on a displacement of 10,000 tonnes or less. What got called a battlecruiser developed over time, with the battlecruiser of 1907 bearing very little resemblance to what was called a battlecruiser in 1918, let alone in 1936. This is the problem with trying to describe the Alaskas as battlecruisers; they meet some of the criteria used to describe some ships as battlecruisers, but not those met by other ships called battlecruisers.

Jacky Fisher's original conception for the ship that would become the battlecruiser was for a large, fast ship armed with battleship guns but armoured like a cruiser. This would attempt to do to the armoured cruiser what Dreadnought had done to the battleship. The envisaged role was the traditional cruiser role - scouting for the battlefleet, defeating the enemy's scouts, supporting the battlefleet when engaged, and countering enemy attempts at commerce raiding. Fisher's colleague and biographer, Admiral Reginald Bacon, wrote

... that ships of the size and tonnage necessary ... should have an additional use in being able to form a fast light squadron to supplement the battleships in action and to worry the ships in van or rear of the enemy's line. They were never intended to engage battleships single handed (italics in original); but they were designed to assist in a general action by engaging some of the enemy's ships which were already fighting our battleships.

The ships built to this concept would be referred to as large armoured cruisers until November 1911, when the term battlecruiser was coined.

As the First World War drew to a close, the battlecruiser started to develop significantly. It starts to resemble less a cruiser with battleship guns, and more a fast battleship. This is true for British and Japanese designs, but somewhat less true for American ones. The first such design was the famous Hood. While she was famously under-armoured, this was only true for her deck armour, and such flaws were common amongst ships of similar vintage. Armour made up 33% of her displacement - the 1936 King George V class battleship had 36% of her displacement as armour - and her belt was thicker than most contemporary battleship designs. The Japanese would modernise their Kongo-class battlecruisers, adding more armour. Some of their never-built designs, such as the cancelled/converted to carrier Amagi-class, had armour comparable to that of later battleships. The post-war G3 design, also cancelled due to the Washington Treaty of 1923, exemplifies this concept of the battlecruiser. They would have had thicker belt and deck armour than any battleship design bar the Yamato class, which had a 14,000 ton advantage on them. Such battlecruisers were not expected to fulfil the classic cruiser role as their predecessors did. Instead, they were expected to, and did, form a significant part of the battleline.

In the 1930s, the battlecruiser label also starts to be applied to ships that don't quite fit into the battleship-cruiser spectrum. These are ships like the German Scharnhorsts and French Dunkerques. These are fast ships, larger and much more heavily armoured than the typical cruiser. But they don't have the heavy armament used by battleships in this period, opting instead for 11-13in guns. By the standards of 1914-18, they'd be battleships - they've got the armour and the armament. But in 1936, they're too lightly armed to fight a battleship, even if they are well armoured. Instead, they fill a role closer to the traditional cruiser one - they're used for commerce-raiding and counter-commerce-raiding operations.

So, how do the Alaska-class fit into this? They definitely don't fit the definition of battlecruiser used by the contemporary RN and IJN. They're not well armed enough to serve as a central part of the battleline as ships like Hood, Repulse or Kongo did. They somewhat fill the last definition, but are somewhat underarmoured compared to the Scharnhorsts. However, if you'd shown them to Jacky Fisher, he'd recognise them as almost the platonic ideal of what he wanted to design. They're a large, fast cruiser, that can outgun any other cruiser afloat. They're designed as a cruiser killer, but were capable of supporting the battlefleet where necessary. Even their designation as large cruisers echoes the designation of the original battlecruisers as large armoured cruisers. Unfortunately for any hope of a simple resolution, this definition of battlecruiser had become obsolete by the time they were launched. This is why the USN didn't classify them as a battlecruiser - in 1944, the term has connotations that the Alaskas definitely don't fulfil.

Sources:

The Grand Fleet: Warship Design and Development 1906-1922, David K Brown, Seaforth, 2010

Nelson to Vanguard: Warship Design and Development 1923-1945, D. K. Brown, Seaforth, 2012

Warships After Washington: The Development of the Five Major Fleets 1922-1930, John Jordan, Seaforth, 2011

The British Battleship 1906-1946, Norman Friedman, Seaforth, 2015

Fighting the Great War at Sea: Strategy, Tactics and Technology, Norman Friedman, Seaforth, 2014

Jane's Battleships of the 20th Century, Bernard Ireland, Collins-Janes, 1996

As a follow-up question:

Given that many nations seemed to be interested in developing some kind of large cruiser between the treaty-defined 8" / 10000 and de facto battleship categories, why didn't we see many actually fielded?

Battlecruisers are not currrently in the game

Kongou is ingame.

Edit: Amagi is also.

No.
A few reasons:
Battlecruisers were significantly faster than contemporary battleships.
Battlecruisers carried armament on par with contemporary battleships.
Battlecruisers can outrun most contemporary cruisers.

The Alaska class fails to meet any criteria.