There seems to be quite a bit of confusion regarding splitting and how different wallets (accounts) work now that we have an ETH chain and an ETC chain. This should hopefully help clear some things up.
First thing you need to understand is that every address already "exists". When you "create a wallet" you are not actually doing anything on the blockchain or communicating with the blockchain in any way.
Creating a wallet means you get a randomly generated private key and the address (public key) that corresponds to that private key. Then you save those two bits of information, along with the password if the private key is encrypted, and decide to use it.
Now that you understand that you will understand that any private / public keypair on the ETC chain is equally valid on the ETH chain and vice versa. They are both derived using the same methods and therefore equally valid, as the ETC chain and the ETH chain used to be a single chain.
So, in the simplest of terms,
What really matters is what funds are in your wallet and what you can do with those funds.
All transactions are also equally valid, just like your wallet, on both chains. If you send an ETH transaction it will be replayed and it will attempt to send on the ETC chain. What does this mean?
An example:
Since it is unlikely that you want to do this (unless you are sending to Poloniex or Kraken where they accept both your ETH and ETC and credit both to your account) you will want to create a wallet specifically for ETH and specifically for ETC. You will not want to hold ETH in your "ETC address" or vice versa. To send Only ETH or Only ETC you need to use the ReplaySafe contract.
This is implemented on MyEtherWallet here via the "Only ETH" and "Only ETC" methods. Instructions for splitting in Mist are here: https://steemit.com/ethereum/@pauls/ethereum-fork-step-by-step-guide-to-safely-splitting-your-eth-etc
My recommendation on how to split via MyEtherWallet.com to two new offline wallets is as follows:
Good luck and hope this clears things up for you!
