Effective Lennard-Jones models for the water-carbon interaction are derived from existing high-level ab initio calculations of water adsorbed on graphene models. The resulting potential energy well (ε_CO+2ε_CH≈1 kJ/mol) is deeper than most of the previously used values in the literature on water in carbon nanotubes (CNTs).Moreover, a substantial anisotropy of the water-carbon interaction (ε_CO≈2ε_CH) is obtained, which is neglected in most of that literature. We systematically investigate the effect of this anisotropy on structure and dynamics of TIP5P water confined in narrow, single-walled CNTs by means of molecular dynamics simulations for T=300 K. While for isotropic models water usually forms one-dimensional, ordered chains inside (6,6) CNTs, we find frequent chain ruptures in simulations with medium to strongly anisotropic potentials. Here, the water molecules tend to form denser clusters displaying liquid-like behavior, allowing for self-diffusion along the CNT axis, contrary to all previous simulations employing spherical (ε_CH=0) interaction models. For (7,7) CNTs we observe structures close to trigonal, helical ice nanotubes which exhibit a non-monotonous dependence on the anisotropy of the water-carbon interaction. Both for vanishing and for large values of ε_CH we find increased fluctuations leading to a more liquid-like behaviour, with enhanced axial diffusion. In contrast, structure and dynamics of water inside (8,8) CNTs are found to be almost independent of the anisotropy of the underlying potential, which is attributed to the higher stability of the non-helical fivefold water prisms. We predict this situation to prevail also for larger CNTs, as the influence of the water-water interaction dominates over that of the water-carbon interaction.