you just put a backup of your wallet there, for security purpose
Maybe you didn't phrase this correctly. If yes, please correct it. This may confuse other users.
A back up in a place never will be a cold storage. What you store in cold storage must not be in a hot/online storage. If yes, then there isn't a need for cold storage because cold storage is protect your Bitcoins from hackings and all but if you store this in a hot/online wallet, private keys are already exposed to danger.
actually i store both, a backup and a different wallet in a cold storage, the first even if you use it in a wallet(low amount) you need a copy of it anyway, your hdd could have be damage one day or other accident, the second with more bitcoin, which it isn't shared with the hot wallet, as you said
But a real cold wallet must be never connected on Internet and especially their copy/buckup should be never exposed outside from that "offline machine". The cold wallet question bring to my mind the Bter cone 'hack', they have said someone stolen 7'000 bitcoin from their cold storage but the unique way for hack and stole those Bitcoins was to be physically in front of that machine/device. Remember if you expose a cold wallet online also for 1 minute you can't continue to name it 'cold'