As in your other question, the fourth proposition is a tautology. The other three propositions can be expressed as
$$
((\neg W_1 \land \neg W_2) \implies \neg Y)
\land
((\neg W_1 \land W_2) \implies Y)
\land
((W_1 \land \neg W_2) \implies \neg Y)
$$
More simply, combine your first and third propositions and omit the fourth one to obtain
$$
(\neg W_2 \implies \neg Y)
\land
((\neg W_1 \land W_2) \implies Y)
$$
Now rewrite in conjunctive normal form, by replacing $P \implies Q$ with $\neg P \lor Q$, pushing $\neg$ inwards, and distributing $\lor$ over $\land$:
$$
(W_2 \lor \neg Y) \land ((W_1 \lor \neg W_2) \lor Y)\\
(W_2 + (1 - Y) \ge 1) \land (W_1 + (1 - W_2) + Y \ge 1)\\
(W_2 \ge Y) \land (W_1 + Y \ge W_2)
$$
Several other examples are here.
A good reference for this is:
Raman, R. and I.E. Grossmann, Relation Between MILP Modelling and Logical Inference for Chemical Process Synthesis, Computers Chem. Engng. 15 (1991).