We study both theoretically and experimentally the free induction decay (FID) of the electron spin associated with a single nitrogen-vacancy defect in high purity diamond, where the main source of decoherence is the hyperfine interaction with a bath of $^{13}$C nuclear spins. In particular, we report a systematic study of the FID signal as a function of the strength of a magnetic field oriented along the symmetry axis of the defect. On average, an increment of the coherence time by a factor of $\sqrt{5/2}$ is observed at high magnetic field in diamond samples with a natural abundance of $^{13}$C nuclear spins, in agreement with numerical simulations and theoretical studies. Further theoretical analysis shows that this enhancement is independent of the concentration of nuclear spin impurities. By dividing the nuclear spin bath into shells and cones, we theoretically identify which nuclear spins are responsible for the observed dynamics. ; Comment: 16 pages, 7 figures