The remaining really insolvent banks are closed, sold and the deposit insurance is provided by the EU as originally planned.
This was never planned in the first place. Deposit insurance was always the responsibility of the member states. The EU has a rule to say that member states have to provide depositor insurance, but the EU doesn't have either the responsibility or the resources to do it itself. Maybe this should have been set up when they created the Euro, but it wasn't.
The EU and the IMF are involved here because the Cypriot government can't afford to pay the deposit insurance of the banks that would go bust immediately, or the currently solvent banks that would go bust as part of the fall-out from the first lot of depositors losing their money. (See below):
Now with all of this talk of bank account confiscation ALL BETS ARE OFF. Even the partially solvent banks are subject to bank runs. What were they thinking?
1) You're a successful business with a loan from Solvent Bank S. You've got a large order from a Customer C with its money in Insolvent Bank I. Insolvent Bank I goes bankrupt, so does Customer C. You can no longer repay your loan to Solvent Bank S. You go bankrupt, Solvent Bank S lose their loan to you, and you lay off your employees, many of whom have mortgages with Solvent Bank S. Solvent Bank S is no longer solvent.
2) You've got your money in Solvent Bank S. Your friends who had their money in Insolvent Bank I just lost everything above 100,000 Euros (or worse, if the government already can't afford to honour its guarantee). Maybe (1) above won't happen, but maybe it will. Do you still keep your money in Solvent Bank S?
Upshot: Even the partially solvent banks are subject to bank runs. Sometimes there are no good options. (*)
(*) Actually there was a good option here, which is that German taxpayers bend over again and cough up enough money to bail everybody out, including the Russian mafia. That would probably have been in their best interests. But it's understandable that their government wasn't going for it with an election coming up...