Small lathe induction motor - off-grid alternatives to capacitor start or modifying SP PH for capacitor
The vacuum cleaner and jigsaw probably have universal motors. They are essentially the same as series DC motors and can be expected to work ok with light-dimmers and even better with cheap speed controllers. A cheap speed controller will likely have a variable DC output. A light dimmer could have either an AC or DC output.

It is likely that the motor with a capacitor in the end cover is a permanent-split-capacitor type of motor. The capacitor has less capacitance than the motor with the external capacitor and it is connected all the time. The motor with the external capacitor is likely a capacitor-start motor. The capacitor is disconnected by a centrifugal switch or some other device once the motor reaches full speed. The starting torque capability is higher, but it takes more current to start.

A cheap speed controller or dimmer that has an AC output might be able to get the motor started without overloading the generator. The right value resistor might also. If it works, it may take so long to get the lathe up to speed that the capacitor or the winding it is connected to could overheat. There is a VFD that claims to be able to do the job, but I believe the price may be higher than a new motor and controller combined and it may not work. There may be only the one manufacturer making something like that.

You might be able to find a 1/4 Hp universal motor. That would probably work without a controller and you could add one if needed.