TOGA9's post is exactly on point.
There are policies that will cover the CFI for flight instruction in an owner's aircraft, but without a special endorsement doing so, the typical policy does not cover an instructor giving instruction for hire.
"Named" is one of the more confusing pieces. A lot of people =think= it means the CFI is covered when giving instruction. It doesn't.
If you do get "on" the owner's policy in some way, you need to sit down and have someone who understands aviation policies ensure that you got what you think you did. Unlike auto insurance, aviation insurance is neither mandatory nor standardized.
A CFI's best bet is always his own policy.
"Who will the liability fall upon?" It depends. The aircraft didn't do it by itself, either you or the "student" did it. In most cases involving a certificated pilot and an instructor, the best answer is probably "both."