For the 20th Century (even with the aberration f the Washington Naval Treaty years) a handy Rule-of-Thumb is that Cruisers have main batteries under 10"/250mm on caliber. Battle-cruisers and Battleships have larger main batteries, typically starting about 12"/300mm.
"Light" versus "heavy" cruisers split at the 7"/175mm dimensions; smaller being "light", and larger being "heavy." USN dimensions were 6" for CL's, and 8" for CA's.
To split the Battle-cruisers (some folk do not hyphenate, and use "Battle Cruiser" to reflect the "BC" designation some use) from BB's, one can look at the armor belt thickness. If the armor belt thickness approximates the main battery caliber, it's safe to call the vessel a battleship. If it's closer to ha;f the caliber, it's probably a BC.
All that being said, the correct term is what ever the 'owners' call it. We saw this in the Cold War times when USN called ships smaller than Destroyers "frigates." While at the same time, the Royal Navy was calling vessels larger than DD's but smaller than CL's "frigates." This water was muddied even further when the navies created dedicated (mostly) "flagship" versions of Destroyer and Cruisers and called them "Leaders." Since these were ship classes of their own, it meant the "Leaders" were mismatched to the ships they were to "lead."
The closes we have come to commonality was with ships our good Professor mentions above. Where the stock Spru-can destroyer hull had one more hull section added (mostly, sorta) to make into a Ticonderoga-class cruiser. This made the Tico's very much a "destroyer leader" in having a similar suite of sensors and weapons. All of which changed when we needed to add Aegis to our cruiser fleet to better serve Carrier Battle Groups
Or some such thing.