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We present a sample of 14 hydrogen-rich superluminous supernovae (SLSNe II) from the Zwicky Transient Facility (ZTF) between 2018 and 2020. We include all classified SLSNe with peaks $M_{g}<-20$ mag and with observed \emph{broad} but not narrow Balmer emission, corresponding to roughly 20 per cent of all hydrogen-rich SLSNe in ZTF phase I. We examine the light curves and spectra of SLSNe II and attempt to constrain their power source using light-curve models. The brightest events are photometrically and spectroscopically similar to the prototypical SN 2008es, while others are found spectroscopically more reminiscent of non-superluminous SNe II, especially SNe II-L. $^{56}$Ni decay as the primary power source is ruled out. Light-curve models generally cannot distinguish between circumstellar interaction (CSI) and a magnetar central engine, but an excess of ultraviolet (UV) emission signifying CSI is seen in most of the SNe with UV data, at a wide range of photometric properties. Simultaneously, the broad H$α$ profiles of the brightest SLSNe II can be explained through electron scattering in a symmetric circumstellar medium (CSM). In other SLSNe II without narrow lines, the CSM may be confined and wholly overrun by the ejecta. CSI, possibly involving mass lost in recent eruptions, is implied to be the dominant power source in most SLSNe II, and the diversity in properties is likely the result of different mass loss histories. Based on their radiated energy, an additional power source may be required for the brightest SLSNe II, however -- possibly a central engine combined with CSI.