Equivalence of Definitions of Distributive Lattice

1 is equivalent to 1'
By applying Meet is Commutative several times, we have:

which (after renaming variables as appropriate) establishes the equivalence.

2 is equivalent to 2'
By applying Join is Commutative several times, we have:

which (after renaming variables as appropriate) establishes the equivalence.

1 implies 2
Suppose that $(1)$ holds, and hence $(1')$ as well.

2 implies 1
By inspection, aided by Dual Pairs (Order Theory), we see that $(2)$ is dual to $(1)$.

Thus by Global Duality, $(2)$ implies $(1)$ as soon as $(1)$ implies $(2)$.

That direction was already established above.

Also see

 * Definition:Distributive Lattice