  Motivated by the on-going discussion on the nature of magnetism in the
quantum Ising chain CoNb$_2$O$_6$, we present a first-principles-based analysis
of its exchange interactions by applying an \textit{ab initio} approach with
additional modelling that accounts for various drawbacks of a purely density
functional theory ansatz. With this method we are able to extract and
understand the origin of the magnetic couplings under inclusion of all
symmetry-allowed terms, and to resolve the conflicting model descriptions in
CoNb$_2$O$_6$. We find that the twisted Kitaev chain and the transverse-field
ferromagnetic Ising chain views are mutually compatible, although additional
off-diagonal exchanges are necessary to provide a complete picture. We show
that the dominant exchange interaction is a ligand-centered exchange process -
involving the $e_g$ electrons -, which is rendered anisotropic by the
low-symmetry crystal fields environments in CoNb$_2$O$_6$, giving rise to the
dominant Ising exchange, while the smaller bond-dependent anisotropies are
found to originate from $d-d$ kinetic exchange processes involving the $t_{2g}$
electrons. We demonstrate the validity of our approach by comparing the
predictions of the obtained low-energy model to measured THz and inelastic
neutron scattering spectra.
