Cold storage wallet is a place to sign the transactions, you can call it a "transaction signing tool" instead of a wallet. you give it the transaction and tell the wallet:
"this (these) output(s) belong to this key and i have checked it while i was online, now sign the transaction with the private key belonging to the public key i gave you".
and that is all the cold storage needs to know
- which key to use for signing
- and what to sign
there can be some very complicate (not at all recommended) ways of somehow forcing the cold storage to show the balance but it is not a good idea because you can simply contaminate the cold storage.
remember a cold storage is secure as long as it has no contact with outside world
(they call it air tight and for extra measures you can even use QR codes to give the wallet the raw transaction to sign and then use QR code to get the signed tx out without even connecting an storage device to the cold storage to get the tx in and out).