Is such a statement theisitic, atheistic, agnostic, or other at the foundation? Is it possible for it to be more than one of these?
Already answered. It's not specifically
anything at its foundation other than common-sense prioritization of questions and answers for discerning truth. It can be said to be theist, atheist, agnostic, or anything depending upon who makes proper use of it or who rejects it/uses it improperly. (Not sure how that can be improperly used, exactly, but allowing for the possibility.)
To what extent does the statement rely upon the belief in absolute Truth or absolute laws of logic?
Well, it obviously assumes that what we experience with our senses is real. If our experiences were illusory, then the idea would fall apart completely. It's also only reliable if truth is absolute, i.e. two opposing truths can't be true at once, things can't be objectively true for one person/objectively false for another; and the absolute laws of logic are essential.
If it doesn't reply upon either of these absolutes, is the statement itself subjective in nature?
Probably. If those two things aren't true, I see little reason to distinguish between "objective" and "subjective," in any case.