Sy-superfly

Are you sure it was white not light blue smoke, because blue smoke is oil burning off, and if you put 2lts of oil in that means there was only 0.6ltrs of oil left in it....... You would have seized it.
With the sv's the big pain is that you fill the oil up onto the clutch which takes AGES it get to the bottom I mean if it's cold nearly 3-4 mins to get to see it all. So my advise is next time you fill it up try to only fill up to just past half way if your in a rush. Easiest way to gauge on your own is to lean it on a wall as close as is safe to upright with a rag on the bar end so not to scratch it. but obviously easier with a friend there.
The thing that will happen when you restart it after draining the oil is it will blow shed loads of smoke for a good few mins so don't be concerned by this it is normal after this little boo boo. I also think you would be very unlucky to made the piston rings go, it's more the oil has pushed past them with pressure.

"I'd recommend taking the spark plugs out, and turning the engine over to blow out any excess oil from the combustion chamber." Good idea to do but Please Please put some rag over the spark plug holes when doing this or the garage will be painted oily colour free of charge : ).