Right! Probably the best overall categorization that can be done with ship classes is based on armor and guns, with the idea that any particular genre of ships (light cruiser, etc) should be able to fight on an equal basis its peers, defeat those of inferior class, and escape those of superior class. Of course, this meant that there was a constant effort to increase the capabilities of the chips of any particular class so that it could either catch those of inferior class, or improve armament so that a lighter unit could destroy a superior unit.
A ‘battlecruiser’ was originally called a ‘dreadnought cruiser,’ meaning it has a uniform heavy battery of guns (not mixed armament as previously) of battleship caliber (11" and up), the armor of an armored cruiser, and the speed of a cruiser. The result is a ship that made all armored cruisers obsolete, but which could not really stand up to a battleship (though it was intended that it could escape one).
The old armored cruisers (and pre-dreadnought battleships too) had a mixed armament of a variety of different calibers for engaging surface targets, such as 8", 6", 4" and a bunch of smaller quick firing guns. The problem with this is that it became difficult to hit a target, as the different calibers had different characteristics, so spotting the fall of shot and adjusting fire was tricky, to say the least! Armored cruisers were used as distant station flagships, and ‘ships of the line’ to flesh out the numbers of battleships if necessary, and were also used to hunt down the various forms of light cruiser (and in pre-dreadnought days, there were a lot of different types of cruiser, such as ‘belted,’ ‘protected’, ‘first class,’ etc, etc).
Heavy cruisers really didn’t develop until after WW1, though they have some relationship with the old armored cruisers. The big difference involves armament (which is essentially uniform, like the battlecruisers, but of ‘cruiser’ caliber), which was limited by the various post WW1 treaties to no more than 8", and tonnage, which was also supposed to be limited to 10,000 tons. Their mission was to operate as screening and scouting units, with enough power to push aside or drive off the scouting and screening elements of opposing forces, and have enough speed to escape battleships. A few ships managed to dodge the restrictions of the treaties, either because they were developed/built before the treaty rules were finalised, or in one way or another could not really be classified under the existing rules (‘pocket battleships’).
Light cruisers more closely embodied the ideal of a ship designed for scouting missions for the main fleet, with light armor and guns (no more than 6") and high speed. It could defeat a destroyer, and was often used by many navies as a ‘destroyer leader,’ or destroyer squadron flagship. As these ships were often as large as a heavy cruiser (though not as well armed or armored), they often worked in concert with heavy cruisers, and were also ideal platforms for AA defensive systems, leading eventually to the ‘AA cruiser.’ These were often used as escorts for carriers, etc…