The Lumina Project: The Demographics of Active Galactic Nuclei from Quasars to Little Red Dots at $z\geq 3$
High-redshift active galactic nuclei (AGN) serve as powerful probes of early black-hole growth, galaxy formation, and the evolving intergalactic medium (IGM). In this work, we use Lumina, a cosmological radiation-hydrodynamic simulation spanning the epochs of hydrogen and helium reionization, which combines a large $(500\,{\rm cMpc})^3$ volume with $2\times 6000^3$ resolution elements, to explore high-redshift AGN. The simulation self-consistently follows hundreds of millions of galaxies and supermassive black holes (SMBHs), together with their impact on the ionization and thermal state of the IGM. We exploit this uniquely large dynamic range to predict multi-band AGN luminosity functions (LFs) at $z \geq 3$, from hard X-rays to the mid-infrared. These predictions encompass both moderately luminous quasars and the faint ``Little Red Dots'' (LRDs) uncovered by JWST. We develop an empirical model that maps simulated SMBHs onto observed AGN using bolometric and extinction/absorption corrections for canonical AGN and LRDs, and in which SMBHs with $M_{\rm BH}\leq 10\,M_{\rm seed} \sim 10^{7}\,{\rm M}_{\odot}$ stay in the LRD phase with a duty cycle of $30\%$. This simple framework reproduces the observed LFs and clustering of LRDs. Meanwhile, the pre-JWST quasar LF constraints are recovered, although we find that a $\sim 0.3$ dex log-normal scatter in bolometric luminosity is required to reproduce the bright end. We place the simulated AGN population in the cosmological context by quantifying the redshift evolution of AGN and LRD number densities, and their contributions to the integrated BH mass densities. The same AGN population is the dominant driver for the HeII reionization modelled self-consistently in Lumina. This empirical AGN model paves the way for general population-synthesis models of high-redshift AGN, including LRDs, in a unified cosmological framework.