This is an extremely common phenomenon and there's nothing you can do to prevent it entirely.
You seem to be well aware of the standard bit of advice going back to the days of slotted screws, that it is vital to press hard to fully engage the driver with the screw. But unfortunately with power driving and Phillips-head screws it is virtually inevitable that you will experience the head slipping out at least some of the time (called 'cam out') which will always cause some damage to the screw, to the screwdriver/driver tip, or quite commonly, both.
Why is it so easy? The ugly fact is that Phillips screws were purpose-designed to cam out!
They were not intended to become consumer-level fasteners but rather were for industrial use, the intent being that on production lines where speed was essential the driver should cam out rather than risk the screw being over-tightened, to prevent them being set too deeply, or snapping. And also to help reduce wear and tear on the power drivers themselves (the motors and gearing, not the driver tips — these are a cheap consumable in industrial settings, literally available to buy in boxes of 100 and more).
Edit: provided by the Comments below, Phillips screws being designed to cam out is apparently a persistent myth, the result of a misunderstanding based on the different use of the term by Phillips in his patent application (Phillips 2,046,837) and in modern usage.
The patent application actually states that the design intention was the exact opposite, particularly in the context of power driving, going so far as to claim:
"Moreover, by reason of the perfect fit between the driver and the screw, the screws may be driven and removed innumerable times without the slightest indication of mutilation to the head."
Now this may be true if both driver and screw have perfectly matching geometries corresponding with the patent but unfortunately in the real world, for whatever reasons, Phillips drivers do cam out very easily and begin to strip the screw (even without a power driver being used).
The problems associated with them led directly to the development of Pozidriv, a linear descendent of the Phillips head designed to have the same benefit (auto-centring) but by providing more facets the tip of the driver engages much better with the screw, and as a result it will cam out far less easily.
So if you need to continue to use Phillips screws buy your driver tips in bulk and simply throw out the stripped ones as soon as they're too worn to work well. But ideally if possible switch to better screws: Pozidriv, Robertson or Torx.