Coupla things...
When you say "starting problems"... what does that mean exactly?
If you are suspecting the coil - we don't know how you've re-done the wiring, but did you check a) the rear stop-light switch and it's bulb, because together they form the ground circuit for the HT side of the coil.
CaptPaul is exactly right.... and in support - without wishing to upset anyone - both ignition coils are exactly that - just plain coils of wire - but wound with different nos. of turns, with different resistances. The stumbling point is that Ducati - or the industry itself chose to name them by the electrical source of the bike to which they're fitted. Duh.
I don't know who here knows what about electrics, and I don't wish to tread on any toes, but - for the total novice - these coils are both basic transformers with two windings - a primary, low voltage winding, and a secondary, high-voltage winding, which is connected to the spark plug.
Assuming a -ve ground, when the points are closed (with the "DC" coil) there is a continuous direct current flowing from the battery negative, through the frame to the points, across the closed points, up the points wire to the coil, through the coil to the ignition switch - and then to the battery positive terminal. (Electron flow is from negative to positive). This flow builds up a whacking great magnetic field in the primary coil, and is maintained until the points open: at this time that current stops, the mag field collapses, and in doing so - it collapses also through the secondary windings, and this induces a current in that winding: however - this winding has many more turns than the primary, so a much greater resistance, and the result of that high resistance (actually called impedance in a coil) is that a greater voltage will rise across the output connections of that secondary winding - somewhere in the 15 - 25 thousand volt range - sufficient to jump across the plug gap in the gasoline-rich atmosphere in the combustion chamber. That collapsing field is forcing the current to travel, and the more windings, the greater the impedance and the higher the voltage. There's lots written on this on the internet.
Bear with me: In the case of the "AC" coil - there is not a solid 6 - 7 (or 12 - 14) volt battery supplying 1-2 amps of current to the "AC" coil - it is getting a varying-frequency (ac) current that is anywhere between perhaps 4 and 40 volts, and at a lower current than that battery can give during a kickstart. This has to mean that the resistance of the AC coil has to be different, the winding resistance changed to match the lower performance of an alternator at kickstart rpms, and so a lower output can be expected (when you're kicking the bike). If you put the "DC" coil on the bike that doesn't have a battery - then you will have "issues". If you put an "AC" coil on a bike with a battery - you might burn the coil out. I have never tried this, and not having a battery ignition coil to measure against an "AC" coil - I can't give a definitive comment.
In the OP's post - it could be the coil, I suppose, but it could be a number of other things, too, such as points, condenser, ign. timing, grounds, etc.. Those things need to be sorted first. To see if it's the coil - with a known good battery connected, the plug sitting on the head, you can just open the points by hand, and you should get a good, fat blue spark. If not - then go digging.
Sorry if I got long-winded.
Cheers,
