The German Navy considered them battleships, while most English-language works consider them battlecruisers. I would argue that they are probably best thought of as similar to the American Alaska-class ships, which the USN considered 'large cruisers'. To understand why, we need definitions of a cruiser, a battleship and a battlecruiser.

The only consistent way to define ship classifications is by the role which they were designed to fulfil; doing so allows us to compare ships between different time periods and different navies (remember that not every navy uses the same classification scheme; a relevant example is that the German Navy only used the term battlecruiser once, to refer to an interwar design that was never laid down). A battleship is easy to define in this way: it is the main fighting unit of the fleet, the heaviest and most important part of the force, and the one that fights the decisive battles. A cruiser, meanwhile, has three main tasks. In no particular order, these are scouting for the battlefleet, protecting merchant ships, and raiding enemy merchant ships. However, some cruisers might be optimised for performing one or two of these roles. The battlecruiser, meanwhile, is primarily intended for operating with the battlefleet. As well as scouting for the battlefleet, like a traditional cruiser, it is also intended to provide the battlefleet with a faster squadron to give commanders more tactical options. As such, the battlecruiser is expected to fight battleships, while the cruiser is not.

The Scharnhorst class were designed as a response to the French construction of the Dunkerque class. Much like the Scharnhorsts, these ships are considered battlecruisers in English-language sources, but were described differently by the French fleet. The prime driver for construction of the Dunkerques was the construction of the German Deutschland class cruisers. These ships were intended as commerce raiders, using their heavy armament to see off the lighter cruisers that Britain and France used to protect their trade. By building a ship that was protected against the 11" guns of the Deutschlands, and with an armament that outweighed them, the French had a potent counter. The scale of the armour and the armament meant that they could also fight the old Italian dreadnoughts, but they would struggle against more modern battleships, and so were prevented from engaging them. The Scharnhorsts were constructed to ensure that the Germans could still maintain a heavy commerce-raiding force, being powerful enough to overcome a Dunkerque, and were almost solely deployed in this role. Battlecruisers would be expected to be able to fight battleships, especially when they outnumbered the opposition two-to-one. The Scharnhorsts refused several such engagements, disengaging from Repulse at the Lofoten Islands in 1940, and avoiding separate engagements with the old battleships Ramillies and Malaya during Operation Berlin. The Kriegsmarine was not unwilling to let its heavy units engage ships of comparable capability in order to attack a convoy; Berwick's engagement with Admiral Hipper on Christmas Day 1940, the Battle of the Denmark Strait or the Battle of the Barents Sea all saw Kriegsmarine units engaging British forces of equal strength. The fact that the Scharnhorsts were prevented from engaging battleships shows that they were not seen as able to do so successfully. They were designed and built to kill treaty cruisers, and other treaty cruiser killers like the Dunkerques. The American Alaskas were built for a similar role, and were considered 'large cruisers' by the USN; this classification implies that they are more capable than the typical cruiser, but are not intended to fight battleships like a battlecruiser does. This describes the Scharnhorsts well, and as such, I'd consider that this is the best term to describe them.