Yes the first one, if the distinction needs to be made.
I'll even go further and claim that distinction itself is meaningless and only matters when say organizing university departments.
What needs to be kept in mind is emergence. Chemistry emerges from physics, then biology from chemistry, ecology from biology, etc. Experimentally verified God will become the base level out of which physics emerges from in the other direction. In that sense, physics' domain perhaps includes everything. But it all depends on how we define the terms.
> the purpose of the intellect is the attainment of truth
This I wholeheartedly agree. Maybe you'll enjoy reading this blog post [0] that proposes Truthism as life's goal.
[0] https://waitbutwhy.com/2014/10/religion-for-the-nonreligious...
But quite a few atheists say that claims about anything that is outside science-in-the-second-sense's domain is irrational. Example: Alex Rosenberg states that any knowledge outside of physics' domain is irrational (in line with his reductionist philosophy, he holds chemistry, biology, etc to be physics on a bigger scale). Hume also seemed to be pushing such views with his fork. This claim is trivially easy to refute, I assume you know the arguments already so I won't waste your time with them.