This is true, in that case your coins will be stored in new wallet on new address which is controlled by new private key, which means that old address is empty and using that old private keys does not pose a security risk for your BTC.

Of course you should pay attention on that new wallet for BTC, always at least double check that it is legal source for download. In case of Electrum BTC wallet use only https://electrum.org/#home and read about posted warning.