And how can this be done safely and surely without the transaction being accepted on the other chain also?
To "split" your coins safely, assuming you have a trusted environment and BTC wallet in the first place:
- Create an completely new BTC wallet, i.e. not sharing any keys with the old one
- After the fork, move all your BTC from the old wallet to the new wallet
- Once your old wallet is empty (on the BTC chain), export its private keys
- Create a new "Bitcoin Cash" wallet, import private keys from your old BTC wallet
Now about the transaction not being accepted on the other chain:
The rationale behind replay protection is basically making transactions incompatible with the other chain by design, i.e. by consensus rules. If someone (or your wallet/client/whatever) records and broadcasts your transaction to the other chain, nodes will reject the transaction. You can read the UAHF specs that I linked in my previous post for the gory details.