Because if it's my guide above, the online watching-only wallet will naturally show it as "watch only" since it's that type of wallet.
Your private keys should be in your offline machine that can be used to sign the transactions created by the watching-only wallet.
If you followed it correctly, you can do the final step "[To Create Transactions]" whenever you need to spend your bitcoins.
I assume LoyceV is right with the watch only balance being connected to the Public Key address, otherwise there's no reason for it to show on block explorer but not on your private wallet. Follow his steps and give us an update so we can properly help you if in any case something happens.
I'm not entirely sure how exactly this would work in Bitcoin Core, so bear with me while guessing: you mentioned the private key isn't in the standard Wif format. What format is it? Is it in Hex? Could it be you converted it to Wif, but got the compressed version, while your 2010-transaction uses the uncompressed Wif? If that's the case, get the uncompressed private key (Bitaddress, offline, you know the drill by now) and import that one into Bitcoin Core.