This is exactly what’s going on: the spill is a cone, and conic sections by a plane not containing the vertex will never be linear.
The difficulty in the alignment method is that, if you angle your target plane in such a way that its extension contains the emitter, the plane occupies zero solid angle from the perspective of the emitter, and thus receives zero light!
The best method I have for calculating spill angle is just to measure the height+width of the reflector’s inner surface, and do a tiny bit of trig.