The proof I'm trying to understand. I don't get why $k$ is unique.
When trying to find the equation of a plane, suppose we're given a normal vector $n=(x_o,y_o,z_o)$ and a point on the plane $p^*=(x^*,y^*,z^*)$. Then the set of all points on the plane are those points $(x,y,z)$ that satisfy $((x,y,z)-(x^*,y^*,z^*))\cdot(x_o,y_o,z_o)=0$.
Using dot product rules, $(x,y,z)\cdot(x_o,y_o,z_o)=(x^*,y^*,z^*)\cdot(x_o,y_o,z_o)$.
But $p^*=(x^*,y^*,z^*)$ can be any point on the plane, and using two different points would change the right hand side of the equation (i.e. two different $k$ values for the same plane).
Help?
No, using two different points in the plane will not change the right hand side, for both points will satisfy the equation of the plane. Think about it.
To be more precise, adopting your notation then if $(x',y',z')$ is another point in the plane and you consider using it instead of $(x^*,y^*,z^*)$ then by the equation of the plane, $(x',y',z')\cdot(x_o,y_o,z_o)=(x^*,y^*,z^*)\cdot(x_o,y_o,z_o)$. The left hand side of this equation is the right hand side of your “new” equation.