I don’t think the delineation between old and new will be quite so clear. Consider a new client that all the miners switch to, importing your wallet to this client causes a transaction to appear on the network transferring all your old coins to a new client, when confirmed all your old coins are now bitcoin2, which can’t be sent to bitcoin1 wallets. ANy attempt to use your old bitcoin1 coins will show up as invalid.
I don’t think the delineation between old and new will be quite so clear. Consider a new client that all the miners switch to, importing your wallet to this client causes a transaction to appear on the network transferring all your old coins to a new client, when confirmed all your old coins are now bitcoin2, which can’t be sent to bitcoin1 wallets. ANy attempt to use your old bitcoin1 coins will show up as invalid.