Battlecruisers inside the cruiser spectrum

Is a battlecruiser really a cruiser?

Yes, in one sense. Battlecruisers occupy the heavy end of a broad cruiser spectrum and are heavily armed ships with armor intended for a particular role.

A battlecruiser belongs to the broad cruiser spectrum. It is heavily armed, with protection judged adequate for its intended role. That role matters when assessing the armor. Ships often end up doing jobs for which they were not designed.