To classify a disjunction accurately, ask two separate questions.
Question 1: Can the alternatives both be true or both be false? This establishes genuine disjunction, exclusion of joint truth, or exclusion of joint falsehood.
Question 2: Does the relevant exclusion follow from the meanings, or from contingent circumstances? This establishes oppositional or coincidental disjunction.
“An integer is even or odd” is both genuine and oppositional.
“This is a tree or a stone” is, in the stated sense, both exclusion-of-joint-truth and oppositional.
Answering both questions yields a more complete analysis of the same proposition.