"save" saves the transaction you just made in your wallet file on the computer you just created it. in other words it will add that transaction to your history as a "local transaction".
"export" exports the transaction to a file with a .txn extension which is basically a JSON file containing your raw transaction. then you can use this file to transfer elsewhere.

i have not used the multisig features of Electrum that much but based on what you are explaining i suppose there is only two possibilities the most possible one is that you imported both keys in your offline wallet and now are signing with both but think you are signing with one because the network will reject your tx as invalid if it was otherwise (signed with 1 but needing 2).
when creating the cold wallet did you give it 1 master private key (xprv) + 1 master public key (xpub)?

the other possibility is that your wallet may not even be a multisignature one.