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Some wallets (such as inputs.io) claim to provide off-chain transactions. Note: I have not used that wallet yet.
Say if I have a Bitcoin address A in such a wallet provider and send 1 Bitcoin to address B from the same wallet provider, what will show on the public blockchain?
There will have to be a public record of a transfer into address B before any transfers can come from address B. Internally the provider may move the funds from one address to the other "off-chain" for purposes of display but until some BTC moves into address B publicly it can't be spent from B.
Most likely when you attempt to send BTC externally from address B, that amount will either be:
1) transferred publicly from A to B and subsequently from B onward (incurring 2 transactions and twice the confirmation delay)
2) transferred to a provider address from address A and be internally deducted from address B inside the provider system and actually transferred from some other provider address
3) publicly paid directly from address A but shown as deducted from B in the provider system
Off-chain transfers are one way to add additional anonymity to Bitcoin transactions but at some point the source address has to be publicly credited with BTC for the exchange to be valid in the block chain.
If this is all managed off-chain within the same wallet provider then there would be no record of the transaction at all in the public blockchain.
It would just be bit-flipping inside of the wallet provider's internal database (just like with a traditional bank, or with PayPal).
For example, if you use a inputs.io wallet (as you said) to pay a merchant who uses inputs.io as a payment processor, there is no point actually sending a tx to the Bitcoin network, when inputs.io can just internally shift who owns what amount of bitcoins. This is reflective of the typical "exchange" model of off chain transactions, where you're dealing with two parties who both have accounts with the same intermediary entity.
This is happening for saving time rather than waiting for miners to confirm a valid transaction in Blockchain.
On the other hand this procedure is not transparent because the transferring of funds it is only happening in a local-based off chain internal ledger.
