Use $\sum 2^{-n}(\|x-y\|_n \wedge 1)$ for the distance on $X$ and $\sum 2^{-n}(\|x-y\|_n \wedge 2)$ for the distance on $Y$.

Use $\sum 2^{-n}(\|x-y\|_n \wedge 1)$ for the distance on $Y$ and $\sum 2^{-n}(\|x-y\|_n \wedge 2)$ for the distance on $X$.