  We study neutral- and charged-current (anti)neutrino-induced dissociation of
the deuteron at energies from threshold up to 150 MeV by employing potentials,
as well as one- and two-body currents, derived in chiral effective field theory
($\chi$EFT). We provide uncertainty estimates from $\chi$EFT truncations of the
electroweak current, dependences on the $\chi$EFT cutoff and variations in the
pool of fit data used to fix the low-energy constants of $\chi$EFT. At 100 MeV
of incident (anti)neutrino energy, these uncertainties amount to about 2-3\%
and are smaller than the sensitivity of the cross sections to the
single-nucleon axial form factor, which amounts to 5\% if one varies the range
of the nucleon axial radius within the bands determined by recent lattice
quantum chromodynamics evaluations and phenomenological extractions. We
conclude that a precise determination of the nucleon axial form factor is
required for a high-precision calculation of the neutrino-deuteron cross
sections at energies higher than 100 MeV. By matching our low-energy $\chi$EFT
results to those of pionless effective field theory ($\cancel{\pi}$EFT), we
provide new constraints for the counterterm $L_{1,A}$ that parameterizes the
strength of the axial two-body current in $\cancel{\pi}$EFT. We obtain a value
of $4.9^{+1.9}_{-1.5}\mathrm{fm}^3$ at renormalization scale set to pion mass,
which is compatible with, albeit narrower than, previous experimental
determinations, and comparable to a recent lattice quantum chromodynamics
calculation.
