You are confusing leakage inductance with true winding inductance.

The leakage inductance is a consequence of flux not linking the coil completely. If flux linkage was complete there would be no leakage inductance. The actual windings would have inductance when measured on their own with a bridge or inductance meter, but there would be no inductive drop component in the load current.

I think it may be easier to consider a transformer. A perfect transformer has a winding inductance and the normal to consider is the primary inductance and this determines the magnetising current.

The primary and secondary should be completely coupled and any flux linking one winding will link the other. Any current in the secondary is reflected as a current in the primary in the ratio of the turns. The equivalent circuit would just be an inductor across the lines, simulating the magnetising current.

Now if all the flux doesn't link the windings then you have the effect of a leakage inductance being placed in series with the load ( you still have the shunt primary inductance but we can forget about it for now as it doesn't affect the load current.)

For a 1:1 transformer you can regard this leakage inductance as being in the primary or secondary side. If the ratio is not 1:1 then this is still valid but you need to multiply the value by the square of the turns ratio if you refer it to the primary side. In addition you have the winding resistance causing an internal drop.

The alternator is less easy to follow but basically if all the rotor flux links the stator winding the output is determined by winding resistance as in the axial air gap machines. When part of the flux fails to link the stator winding you have the effect of an added series inductance which we know as the leakage inductance. The final load is determined by the vector impedance comprising the winding resistance and the leakage inductance.

As the inductive bit rises with frequency there usually comes a time with Xl greater than R where the thing goes into constant current mode.

This is very simplified and there are effects of the magnetic field in the stator due to load current reacting upon the flux in the rotor and distorting, reducing or increasing it due to something known as armature reaction.

A capacitive component in the load will increase the effective rotor flux and an inductive load will lower it. Capacitive loads will to some extent compensate the drop due to leakage inductance and adding capacitors to an alternator terminals will reduce the effective volt drop. If you can use large enough series capacitors you can actually neutralise the leakage inductance at a fixed speed but it requires rather large capacitors and is not generally practical.

Hope this helps .