Multiplying by purely complex exponential is basically a phase shift. If you shift original image, the amplitude of its frequency components remains the same, the only thing that changes is the phase of those components. That's the shifting property of an FFT.
Shifting is done by moving the origin point of an image. If $f(x,y)$ is your original image, then $f(x+x_0,y+y_0)$ is your spatially shifted image where point of image originally located at $(0,0)$ is now located at $(x_0,y_0)$.
The first formula is describing something else — the multiplication of an image by sinusoid (complex exponential). It will translate image's components in the frequency domain.