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That first bullet point doesn’t really make sense in the context of the paragraph, which was to define a lattice as an algebraic structure (on bare sets, not posets).
I don’t quite see what you are driving at about the symmetry. Yes, the axioms could interchange $\wedge$ and $\vee$ and you’d get the same thing. So? (Cf. the fact that the notion of lattice is self-dual.) But starting from the algebraic axioms before your edit, the standard partial order $\leq$ is defined through $a \leq b$ iff $a = a \wedge b$, which is equivalent to $b = a \vee b$ by absorption, and this restores the asymmetry.
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