What I do not get is why does the author choose to buy an ITM put. If
the goal was to not lose more than 5.6%, he could have chosen a out of
money put where the strike is ~6% OTM.
The reason why he is buying a ITM put instead of a put 5-6% below the ATM price, is because he wants to only lose 5-6% after all fee's. A put at 5-6% below ATM is not free, so it will not actually provide a 6% cushion, more likely 10%-15% maximum loss after it's cost is accounted for.
You cannot rely on the strike alone to determine the level of protection you are buying.
Real world example. SPY DEC 2017 195 strike put, costs $2150, it's about 6% OTM, but it costs roughly 10% of SPY $207, at best it would protect 85% of your net worth.
Strike - Costs = Protection
Did he choose an ITM put because he does not want to pay any time
premium? Does he not lose in wide bid-ask spreads what he gains by not
paying time premium?
Nope, you were just misunderstanding how he calculated his protection. He wanted to protect 5-6% after the cost of the hedge. He 'needed' to select an ITM put because time premiums are so high that an OTM put wouldn't suffice.