Ride London 100

Whether something penetrates through your tyre is down to force applied to the casing by the object (or vice versa). If the contact area of your tyre with the flint/glass/nail is constant then a higher tyre pressure will apply a higher force to the tip of the flint/glass/nail and you will get more penetration.
Running your tyres at the right pressure will minimise rolling resistance while minimising puncture risk and detrimental handling. Overpressure will definitely increase puncture risk and may also increase rolling resistance for certain scenarios.