Danny Hamilton,
You wrote above: "Private key can be used to sign bitcoin transactions" - HOW, manually? Or the wallet does it for you?
The maths required to sign a transaction would be complex and time-consuming to try to do with pencil and paper. Technically, you could, but it would be a pretty silly thing to do. Generally, people use computer software (such as a Bitcoin wallet) or hardware (such as KeepKey) to build and sign the transaction before broadcasting it.
On my KeepKey, there is a PHYSICAL BUTTON that I have to press to confirm each SEND transaction.
Nowhere in Chrome KeepKey client do I see a field where I need to enter my private key manually!
The private key (or the information needed to generate the private key) is stored in your KeepKey. The client software interacts with the KeepKey so that the transaction can be signed.
Am I understanding it correctly that KeepKey (or ANY "cold" wallet) uses private key IN THE BACKGROUND to sign transactions when I press that button?
Correct. I'm not an expert on the KeepKey system, but if it was created securely, then the client should send the necessary transaction information to the KeepKey device, then the KeepKey should use the internal private key to create the signature when you press the button, then the KeepKey should supply the signature (or signed transaction) back to the client.