I want first to change your notations, sticking to the usual variables $x,\xi$ in the phase space. As a general statement about pseudo-differential operator with a symbol $a(x,\xi)$, I wish to write
$$
\bigl(\text{Op}(a) f\bigr)(x)=\int e^{2iπ x\cdot \xi} a(x,\xi) \hat f(\xi) d\xi.
$$ 
Let $y\in \mathbb R^n$ be given. Then we have 
$$
f(x+y)=\int e^{2iπ (x+y)\cdot \xi}\hat f(\xi) d\xi
=
\int e^{2iπ x\cdot \xi}e^{2iπ y\cdot \xi}\hat f(\xi) d\xi=
\bigl(\text{Op}(a_y) f\bigr)(x), \quad a_y(x,\xi)=e^{2iπ y\cdot \xi},
$$ 
which is your first formula.

Let $\eta\in \mathbb R^n$ be given. Then we have 
$$
e^{2iπ x\cdot \eta}f(x)=\int e^{2iπ x\cdot \xi}e^{2iπ x\cdot \eta}\hat f(\xi) d\xi=\bigl(\text{Op}(b_\eta) f\bigr)(x),\quad b_\eta(x,\xi)=e^{2iπ x\cdot \eta},
$$
and this is your second formula.