  Deciphering the formation of supermassive black holes~(SMBHs) is a key
science goal for upcoming observational facilities. In most theoretical
channels proposed so far, the seed formation depends crucially on local gas
conditions. We systematically characterize the impact of a range of gas based
black hole seeding prescriptions on SMBH populations using cosmological
simulations. Seeds of mass $M_{\mathrm{seed}}\sim 10^3-10^{6}~M_{\odot}/h$ are
placed in halos that exceed critical thresholds for star-forming, metal-poor
gas mass and halo mass (defined as $\tilde{M}_{\mathrm{sf,mp}}$ and
$\tilde{M}_{\mathrm{h}}$, respectively, in units of $M_{\mathrm{seed}}$). We
quantify the impact of these parameters on the properties of $z\geq7$ SMBHs.
Lower seed masses produce much higher BH merger rates (by factors of $\sim10$
and $\sim1000$ at $z\sim7$ and $z\sim15$, respectively). For fixed seed mass,
we find that $\tilde{M}_{\mathrm{h}}$ has the strongest impact on the BH
population at high redshift ($z\gtrsim15$, where a factor of 10 increase in
$\tilde{M}_{\mathrm{h}}$ suppresses merger rates by $\gtrsim 100$). At lower
redshift ($z\lesssim15$), we find that $\tilde{M}_{\mathrm{sf,mp}}$ has a
larger impact on the BH population. Increasing $\tilde{M}_{\mathrm{sf,mp}}$
from $5-150$ suppresses the merger rates by factors of $\sim8$ at $z\sim7-15$.
This suggests that the seeding criteria explored here could leave distinct
imprints on the redshift distribution of LISA merger rates. In contrast, AGN
luminosity functions are much less sensitive to seeding criteria, varying by
factors $\lesssim2-3$ within the seed parameters we have explored. Such
variations will be challenging to probe even with future sensitive instruments
such as Lynx or JWST. Overall, our systematic parameter study provides a useful
benchmark for development of seed models for large-volume cosmological
simulations.
