My approach:
To find $\dim(W1+W2)$, I use the equation $$\dim(W1+W2) = \dim W1 + \dim W2 - \dim(W1 \cap W2)$$
To find $W1 \cap W2$, I simply say that it consists of all the vectors that satisfy both the constraints of $W1$ and $W2$. Hence, I get $\dim = 1$.
Thus, getting $\dim(W1+W2) = 3$. What did I do wrong in the above method?
$$\dim(W_1+W_3) =3$$ is true and you are write that $W_1+W_2 = \mathbb{R}^3$.
A basis of $W_1$ is $\{(1,0,1), (0,1,1)\}$ and note that $(0,2,1) \in W_2$
$$\det\begin{bmatrix}1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 2 & 1 \end{bmatrix}=-1 \ne 0$$
Hence $\{(1,0,1), (0,1,1), (0,2,1)\}$ is a basis of $W_1+W_2 = \mathbb{R}^3$.