We know that (as per the general definition of $\wedge$)
$x\wedge y=x$ in a lattice $(L,\le)$
Again $\wedge $ being commutative in $L$, we have
$x\wedge y=y\wedge x=y$, by definition.
So what we find is that the lub of two distinct elements $x$ and $y$ are different. How is this possible?
So what I am assuming that in a lattice, $x\wedge y\ne y\wedge x$, i.e $\wedge$ is not commutative but this is also contradicting the definition of lattice (or algebraic lattice, since there there is a $1$-$1$ correspondence between lattice ordered set and algebraic lattice).
So what's happening here?
I disagree with your "general definition". In fact, I would say $$ x \wedge y = x\quad\Longleftrightarrow\quad x \le y $$