Both exist currently. There are "on demand" deposits (current accounts, checking accounts or whatever you want to call them) which are considered "money" (and added to physical money to get the M1 money supply). And there are other deposits (like saving accounts) which are investments. The distinction is quite blurred nowadays, you are not getting any interest on them anyway, but if the initiative had passed banks would only keep in their balance sheets the investment-like deposits and not the money-like deposits (which would be transferred to the central bank). Edit: to be clear, current accounts and savings accounts are quite different now from the point of view of the bank: fractional reserve requirements apply only to the former, allowing for "money creation".
So presumably the banks would have offered accounts that were as close as they could be to current accounts while still counting as savings accounts according to the regulators. The impact of the law would have depended on exactly how similar the regulations allow these saving accounts to be to current accounts.
I suspect most people would have moved their money to "savings" accounts and things would continue as before.
It's hard to know how things would have been implemented and how the different parties would have behaved, but I suspect it's clients that would need to be convinced by banks and not regulators.
Taking UBS as an example, the interest rate offered on the current account is 0%. The interest rate offered on the savings account is 0.01% (and it comes with some restrictions, withdrawals above CHF 50'000 p.a. require a three-month notice).
Even if they were otherwise identical, why would anyone put their money in the "risky" savings account instead of the "safe" current account? Interest rates paid to depositors would have to rise to make saving accounts attractive.
