The basic difference between AC and DC Cdi's.......
Both need to charge the internal discharge capacitor to about 200 Volts.
The AC type gets this charging voltage from the poorly assembled and fragile LT coil on the stator plate. You can check this voltage with a multimeter set to the 200Vac range with one lead to earth and the other on the green ignition wire. You should see around 60v at tickover which makes a weak spark to around 200V when revved out, which makes a good spark (not that you will be able to see this of course)
A DC Cdi needs a 12V DC supply. Internally within this CDi the 12V is fed into an electronic inverter, which converts the 12V supply to the required 200V capacitance discharge voltage. The advantage here is that the 200V capacitor discharge voltage is uniform regardless of engine speed making the scoot easy peasy to start, firm regular tickover and resistant to plug fouling. The easiest way to acquire the 12V supply is from the sidelight (rear tail lamp) circuit. Positions 2 and 3 on the key make this line live. Positions 1 and 0 kill it.
There is one caveat here. Position -1 also turns the ignition on but.......... how many of us actually use our parking lights?
By connecting the green ignition wire of the harness to the red wire of the stator, key positions 0 and -1 act as an ignition pickup kill switch. So, if left with the parking lights on, there would be power supplied to the CDi unit but the pickup would remain connected to ground and so the engine will not start. The constant power drain consumed by the CDi in this key position -1 quiescent state, would only be around 2.5Watts. So no big deal really.
But my scoot doesn't have a battery or a battery tray and I don't want to retrofit one..........
This is where the LUkAS style (other brands are available

) comes into play as a compromise between the two. In this case, the CDI derives its power from the low voltage, AC lighting coils. This voltage is rectified and regulated within the CDi before it is sent to the internal 12/200V inverter circuitry. On some machines, a problem can arise where switching the main lights on can kill the ignition on tickover and this is usually due to iffy stator coils or weak magnets on the flywheel.
If AC derived Cdi was better than DC then all modern road bikes would have it. Very few do.