The conditions for the generation of runaway electrons (RAEs) in a magnetically insulated coaxial air diode with needle and conical, with the Taylor opening half-angle of 49.3°, graphite cathodes are compared. The axial magnetic field makes it possible to focus the RAE beam on the collector current probe, thereby increasing the sensitivity of the recording technique used. It has been established that the threshold voltage for RAE generation for the Taylor cone is lower than for the needle (i.e., the cone with a small opening angle), which indicates its non-monotonic angular dependence with a minimum occurring at an angle not exceeding the Taylor one. According to our estimates, the dynamics of free electrons changes qualitatively at the Taylor angle. At larger angles, they accelerate throughout the entire gap; at smaller ones, they accelerate near the cathode and then decelerate at the periphery.