Link to AXA to MT2 tool holder, appears to have oval opening for the arbor tang to prevent rotation.
That's the same as my BXA MT3 Holder. And yes, it has a slot for the MT3 Tang.
With modest pressure, mine deflects 2 thou to the rear side (away from the operator). The deflection is linear with tool pressure.
Using the BXA tool holder, the axis of rotation is offset about 4 inches rearward (away from the operator) from the centerline of the tool post. Everything is rubber. The tool post and tool holder are very solid. But a torque force with a moment arm of 4 inches is not trivial.
On the contrary, a tailstock has zero rearward offset. There is no moment arm and therefore, the deflection is zero.
Note - this is only rearward deflection. Both the tailstock and toolpost will also generate an upward deflection that is the result of the force offset of half the swing of the lathe and the vertical structural strength of the toolpost, tailstock, and lathe bed. But my tailstock and my bed are virtual beasts in comparison to my toolpost and MT3 BXA Toolholder.
Caveats:
1. Every lathe will be different. The smaller the lathe the greater the deflection. The stronger the structure the lower the deflection.
2. Any deflection will be linear with tool pressure. The amount of deflection is dependent on tool pressure.
3. I used a center mounted in the chuck and in the Morse taper and a 5" tool between them. Other geometries will affect measurements.
4. I did not measure tool pressure but tried to be consistent with a pressure about the same as I would apply drilling with a half inch drill.
5. My indicator was mounted on my headstock. Thus, it was also measuring any offset generated by any twisting of the carriage. This is probably minimal but..... Everything is rubber, and that deflection should be included too.