Your lack of comprehension is astounding, but you probably didn't get that.
Granted. I did not do any research before responding. Now I have. Here's my take on it, and while testing it is impossible for me (no PCIII on my 919) it does make sense.
First, a basic assumption: installing a starter / kill switch assembly from a CBR cured the problem. Cool. The question is how did it cure it? First, it's probably a new or newer switch than the one it replaced and the contacts are relatively pristine, guaranteeing a cleaner voltage from the kill switch to the load it supplies. Always a good thing, but maybe not the whole story. Part two is the electrical difference between the two switches: stock, the power to the starter button comes from the output side of the kill switch which will disable the starter when the kill switch is off. The CBR switch does not have this function. Otherwise, they are electrically the same. On the 919 all the kill switch powers is the engine stop relay and the starter relay.
Next, some basics about relays. All conventional relays have two operational voltage ratings. One is pull up voltage, usually listed as the coil voltage. It creates a powerful enough magnetic field to attract the steel armature and make contact. The second rating, rarely specified outside the data sheet, called hold voltage. It is the minimum power required to maintain contact, usually about 60% of the pull up power. This will become important later. Another characteristic is when a relay is deactivated it generates a negative voltage spike, or "back emf" in the same way an ignition coil does which can go as high as -250 volts for a few milliseconds. Generally, the higher the current rating of the contacts the stronger the back emf will be.
Putting it all together, when the starter button is pressed the relay applies battery voltage to the starter motor. The load drops the battery voltage to about 10 volts if it's in good shape and it's warm out. The colder it is the harder the battery has to work to spin the motor, and at the same time the weaker the battery is. Under these conditions the voltage may come close to the hold up voltage of the engine stop relay, but still not low enough to drop it out. That is until the starter button is released, the relay deenergizes, and a negative voltage spike is generated which will be sufficient to arc across the just disconnected starter switch contacts and apply itself to the output side of the kill switch. Under normal circumstances it would be absorbed by the voltage line from the fuse box, but with contaminated contacts in a ten year old switch the bulk of it may apply to the voltage holding the stop relay closed, temporarily deenergizing it. The ECU is "hardened" against this with high value capacitors that keep it alive until the relay reengages. The PC, however, may lose its initialization and cut power to the injectors, killing the motor.
If this is the case, and keep in mind this is all speculation, how to cure the problem economically? Simple. All that needs to be done is eliminate the back EMF from the starter relay by adding a shunting diode to the switched positive line at the relay to ground with the cathode (banded end) connected to the positive line of the coil. It will ignore the starter signal but bypass the negative spike to ground. No spike, no stop relay dropout.
As I said before I can't test this, so if someone with a PC who has the cold start problem kluge this up for a quick test? At most it would cost a dollar or two for a 1N4007 diode and take maybe ten minutes to hook it up. At worst it will have no effect.