You're not moving BTC to "split" your coins... The fork and replay protection had already effectively done that.
You're moving your BTC to protect your BTC from theft should whatever BCH wallet you choose to use turn out to be "bad" and it steals your wallet/seed/private keys...
If you move your BTC to a completely new wallet... Even if ElectronCash then sends your old seed/private keys to the hacker, your BTC is safe as it is in a different seed/private key.
Sending to an exchange will do the same thing, but if you send it back to your old Electrum wallet, you're putting it back on the old seed!
The idea of using a 2nd machine or a VM or live distro CD/USB further increases the security by preventing your BCH wallet from being able to steal your wallet file for new wallet, as it would effectively be on a "separate" device
This is the user who produced electron cash:
https://bitcointalk.org/index.php?action=profile;u=261027I think electron cash should be pretty safe.
You say "If you move your BTC to a completely new wallet".. how can I do that ? I don't have another computer, so is it safe to create a new wallet by opening my Electrum ad use the "New/restore" menu item
?
Yes, make a new wallet on your normal computer.
If it's a large amount of bitcoin, you could always run the separate electrum client on a vps for a fairly low fee (but it should be fairly safe anyway).