Hydrogen-rich Type II supernovae (SNe II) are the most frequently observed class of core-collapse SNe (CCSNe). However, most studies that analyse large samples of SNe II lack events with absolute peak magnitudes brighter than -18.5 mag at rest-frame optical wavelengths. Thanks to modern surveys, the detected number of such luminous SNe II (LSNe II) is growing. There exist several mechanisms that could produce luminous SNe II. The most popular propose either the presence of a central engine (a magnetar gradually spinning down or a black hole accreting fallback material) or the interaction of supernova ejecta with circumstellar material (CSM) that turns kinetic energy into radiation energy. In this work, we study the light curves and spectral series of a small sample of six LSNe II that show peculiarities in their H α profile, to attempt to understand the underlying powering mechanism. We favour an interaction scenario with CSM that is not dense enough to be optically thick to electron scattering on large scales - thus, no narrow emission lines are observed. This conclusion is based on the observed light curve (higher luminosity, fast decline, blue colours) and spectral features (lack of persistent narrow lines, broad H α emission, lack of H α absorption, weak, or non-existent metal lines) together with comparison to other luminous events available in the literature. We add to the growing evidence that transients powered by ejecta-CSM interaction do not necessarily display persistent narrow emission lines.