Re charging, sorta simplified. (This presumes a 12 volt scooter battery. If it's a 6 volt battery, the problem with sticking it on a car's 12 volt charging system should be obvious.)
The charging circuit on most small engines is not designed to put out the number of amps that an automotive charging circuit does.
It is more of a "trickle charge" situation than in a car, which might charge the car's battery at a rate of perhaps forty or fifty amperes of electrical current.
In order to charge a car battery at that high a rate, the charging circuit must allow the alternator/regulator system to generate a high voltage, sometimes as much as 14-15 (or more) volts. That's a "hard" 14-15 volts, where the heavy load of a dead battery will not "suck down" the output voltage of the charging circuit.
But the charging circuit of a small engine does not use much regulation, and is designed to put out a max of about 13.6 volts. This 13.6 volts is exactly the voltage of a freshly charged lead-acid battery. So when the battery is fully charged, almost no current goes into the scooter battery. It does not work like a car regulating circuit, where the battery voltage is actually "sensed" and the regulator shuts down the alternator at 13.6 volts.
Now, if the scooter battery is dead or not fully charged, the charging circuit will not generate the "hard" 14-15 volts that a car charging circuit will, but gets "sucked down" (drops the voltage) to a point just above the dead battery's voltage so that it only "trickle-charges" the scooter battery at a low number of amperes, perhaps only 1 or 2 amps. This charging rate (amps) drops to nearly zero as the battery voltage comes up to that 13.6 volts from being "trickle-charged."
But if you try to jump a dead small engine scooter battery directly to a running car engine, the car will try to pump a lot of current into that battery, and will do so at a high voltage. The circuitry and the small scooter battery are not robust enough to take that kind of charging rate and voltage.
This is especially true if the car was just started, where its own battery was just drained a little from the starting operation, so the car is trying to recharge the big, heavy car's batttery back up to 13.6 volts.
That is why the advice is to charge the scooter battery at first just from the car battery without the engine running, because the maximum voltage of the car battery will only be 13.6 volts tops, and the charging rate will be low.
On the other hand, if you're "jumping" another car, with its big battery and hefty charging circuits, you can do it with the first car's engine running.
Terry, 230RN
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Notes:
The charging and discharging of lead-acid batteries is highly complex and still subject to much study. The 13.6 volts of a freshly and fully-charged battery is sort of nominal and depends a lot on temperature, etc. The same is true of charging voltages in an automobile. I have seen them go up to around 18 volts while charging a "flat" battery.
By "hard," by the way, I mean "low impedance," whereas the scooter charging circuit is "high impedance."
When you say "a 12-volt" battery, that's also a nominal sort of thing. A fully-charged, but "rested" car battery's voltage is actually around 12.6 volts --again, depending on temperature.
The car's alternator can even be tricked to put out 110 volts and there used to be add-ons to allowed you to run 110 volts appliances if they had "universal" motors --like hand drills and saws and the like. I don't know offhand if they still do.
