Observations of hyper-luminous quasars at $z>6$ reveal the rapid growth of supermassive black holes (SMBHs $>10^9 rm M_{odot}$) whose origin is still difficult to explain. Their progenitors may have formed as remnants of massive, metal free stars (light seeds), via stellar collisions (medium-weight seeds) and/or massive gas clouds direct collapse (heavy seeds). In this work we investigate for the first time the relative role of these three seed populations in the formation of $z>6$ SMBHs within an Eddington-limited gas accretion scenario. To this aim, we implement in our semi-analytical data-constrained model a statistical description of the spatial fluctuations of Lyman-Werner (LW) photo-dissociating radiation and of metal/dust enrichment. This allows us to set the physical conditions for BH seeds formation, exploring their relative birth rate in a highly biased region of the Universe at $z>6$. We find that the inclusion of medium-weight seeds does not qualitatively change the growth history of the first SMBHs: although less massive seeds ($<10^3 rm M_odot$) form at a higher rate, the mass growth of a $sim 10^9 rm M_odot$ SMBH at $z<15$ is driven by efficient gas accretion (at a sub-Eddington rate) onto its heavy progenitors ($10^5 rm M_odot$). This conclusion holds independently of the critical level of LW radiation and even when medium-weight seeds are allowed to form in higher metallicity galaxies, via the so-called super-competitive accretion scenario. Our study suggests that the genealogy of $z sim 6$ SMBHs is characterized by a rich variety of BH progenitors, which represent only a small fraction ($< 10 - 20%$) of all the BHs that seed galaxies at $z > 15$.
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