Battleships only exist as mothballed reserves. A battleship was one of the largest ships in the fleet with massive guns and super thick armor. It was meant to fight in the major fleet vs. fleet battles. The name stems form "line of battle ship", because these ships were typically arrayed in a line starting back in the late middle ages. When WW2 showed that airplanes and missiles had made big guns and armor redundant, battleships were relegated to being giant anti-air plattforms that escorted the aircraft carriers. No one built any new ones since then, though "missile battleships" were studied for a while.

A massive step down in size are cruisers, who used to be ships that operated independently form the fleet. There are very few of those around nowadays and they tend to be the core of a fleet instead of cruising alone. You will often see them armed with hundreds of anti-ship missiles nowadays.

The next step down are destroyers, which are still fleet escorts (mostly against aircraft, hence Air Warfare Destroyer) in the major navies and have also taken up the role of going on independent operations. This is because smaller navies tend to not have any destroyers or bigger ships, so sending one destroyer armed with say 64 missiles can be sufficient. A destroyer used to be a relatively small ship that had the job of defending the fleet against other small ships like torpedoboats or submarines. The name stems from "torpedoboat destroyer".

The next smaller categories is frigates. These are the most numerous and are the workhorses of modern navies. They do convoy escort, but also defend bigger fleets from submarines. Many navies also send them on independent missions. They will tend to have 16 or 32 missiles, but won't typically have the big anti air radar a destroyer would. "Frigate" was the name used for sailing ships that couldn't stand in the line of battle. I.e. in say 1750 there would be only two categories: line'o'battleships and frigates.

Those categories are the major ocean-going combat ships. There are lots of smaller ships below that and a lot of support ships, too.

The least manoeuvrable ship of the fleet, typically the flagship of the pre-WW2 battle group with escort of destroyers sometimes also cruisers, frigates and support depending on what it was up to. A role that has been with the carrier in the post-WW2, increasingly missile world. I don't think any navy now retains any battleships, though most kept them around decades past their sell-by date.

The battleship (group) was the power that was used to project, and could be sent to far flung locations to bombard an inland target 25 or 30 miles away from the ship, conduct a naval battle or blockade a port well over the horizon. This is the doctrine of the late Victorian era and lapsed in the inter-war years. WW2 and Pearl Harbor very clearly put power projection with the carrier group. The job of the dozen 16" (18" in WW1, and Japanese WW2) and dozen or more 6" guns has been replaced post-war by air, and increasingly cruise and anti-ship missiles in the arsenal. A modern destroyer gets just 1-4 6" pea shooters with range only around 10 miles, but a decent selection of missiles, submarine and air defence that more than makes up for lack of gunnery.

Heavy shore bombardment is far cheaper by 16" capital guns than by missile, the last real instance being Desert Storm. In this role a battleship has no peer, it's less accurate than missile whilst flinging an awful lot more weight, but more accurate than heavy bombing. The modern era prefers precision by very expensive munitions. :)

...defined in a way in which all the opponents we've fought real wars against in the last 7 decades are not “real opponents”...

The strategic nuclear triad tends to ensure that the opponents you consider “real opponents” are not our real opponents in the real wars that happen in the real world.