Context. Stellar dynamic-based black hole mass measurements of M 87 are twice that determined via ionized gas kinematics; the former are closer to the mass estimated from the diameter of the gravitationally lensed ring around the black hole. Aims. Using a deeper and more comprehensive ionized gas kinematic data set, we aim to better constrain the complex morphology and kinematics of the nuclear ionized gas and thus gain insights into the reasons behind the disagreement between the mass measurements. Methods. We use new narrow field mode with adaptive optics and wide field mode integral field spectroscopic data from the Multi Unit Spectroscopic Explorer instrument on the Very Large Telescope to model the morphology and kinematics of multiple ionized gas emission lines (primarily Hα+[N II] λλ6548,6583 and [O I] λ6300) in the nucleus of M 87. We used Kinemetry to fit the position angle, inclination, and velocities of the subarcsecond ionized gas disk. We used KinMSpy to create simulated datacubes across a range of black hole masses and disk inclinations, and parameterized the differences of the resulting residual (observed minus simulated) velocity maps, in order to obtain the best-fit model. Results. The new deep data set reveals complexities in the nuclear ionized gas kinematics that were not seen in earlier sparse and shallower Hubble Space Telescope spectroscopy. Several ionized gas filaments, some with high flow velocities, can be traced down into the projected sphere of influence. However, not all truly pass close to the black hole. Additionally, we find evidence of a partially filled biconical outflow, aligned with the jet, with radial velocities of up to 400 km s-1. The subarcsecond rotating ionized gas "disk"is well resolved in our datacubes. The velocity isophotes of this disk are twisted, and the position angle of the innermost gas disk (≤5 pc) tends toward a value perpendicular to the radio jet axis. The complexity of the nuclear morphology and kinematics (the mix of a warped disk with spiral arms, large linewidths, strong outflows, and filaments crossing the black hole in projection) precludes the measurement of an accurate black hole mass from the ionized gas kinematics. Two results, each relatively weak but together more convincing, support a high-mass black hole (∼6.0 × 109 M⊙) in a low-inclination disk (i∼25) rather than a low-mass black hole (∼3.5 × 109 M⊙) in a i=42 disk: (a) Kinemetry fits to the subarcsecond disk support inclinations of ∼20-25 rather than 42; and (b) velocity residual (observed minus simulated) maps with slightly smaller residuals are found for the former case. The specific (sub-Keplerian) radiatively inefficient accretion flow (RIAF) model previously proposed to reconcile the mass measurement discrepancy was also tested: the sub-Keplerian factor used in this model is not sufficiently small to make a high-mass black hole in a RIAF inflow masquerade as a low-mass black hole in a Keplerian inflow. In general, Keplerian disk models perform significantly better than the RIAF model when fitting the subarcsecond ionized gas disk. Conclusions. A disk inclination close to 25 for the nuclear gas disk (rather than the previously posited 42) and the warp in the subarcsecond ionized gas disk help reconcile the contradictory nature of key earlier results: (a) the mass discrepancy between stellar and ionized gas black hole masses (our results support the former) and (b) the misorientation between the axes of the ionized gas disk and the jet (we find them to be aligned in both two and three dimensions). Furthermore, we identify a previously unknown 400 km s-1 (partially filled) biconical outflow along the (three-dimensional) jet axis and show that the velocities of the two largest ionized gas filaments at 8"30"nuclear distances can be explained primarily by rotation in the extension of the nuclear ionized gas disk (inclination ∼25).