Astronomy question:

does that total percentage of the moons entire surface (not just the part that faces earth*) being illuminated by the sun at any time depend on the moon phase?

Yes, absolutely

Explanation:

The moon has a total 8 phases and its due to these phases that the sunlight is reflected by the moon that shows its surface boundary when viewed from or on earth. The new phase is completely blank phase as the sun as the moon's disk is completely in sun shadow, and occurs from 6 am to 6 pm. The second phase is waxing crescent that can be seen after post dusk and remains form 9 am to 9 pm. The fort quarter of the moon disk that remains 50% is seen at a 90-degree angle from the 6 pm till noon. The full moon when 100% of the illuminated disc can be seen from 6 pm to 6 am.

Answergranulation;