Google's "Web Mercator" projection is a very different beast from the Mercator projection. The former provides a coordinate system for the area you are viewing, oriented in the way you rotated the view.
The formulas you put in your original question are probably more computationally efficient when used for this purpose, but they do not foster understanding.
In fact, those formulas are obfuscatory, since they include an eccentricity term $e$ (this is not the base of the natural logarithms), but Google uses a spherical model where $e=0$.
The letter $a$ (the one without the subscript) is typically used for an ellipsoid's semimajor axis (for Earth this is about 6378 km depending on the geodetic model used). When assuming a spherical Earth, use $R$ instead, to refer to Earth's "radius".
$a_{map}$, on the other hand, is the map scale, the desired ratio between distances between points on the ground and their corresponding points on the map.
Notice that the equations you provide give derivatives, that is, the rate of change of one parameter based on another. They give $dx/dE$ and $dy/dN$ instead of $x$ and $y$. E and N are the coordinates derived by rotating and aligning the Earth to your view. Because Google assumes a sphere, you can get them by running the latitude and longitude through an azimuthal equidistant projection with the given center and rotation.
The Mercator projection is a cylindrical one, that is, the coordinate system is developed by wrapping a cylinder around the Earth, projecting its features onto the cylinder, and then unrolling the cylinder to lie flat.
This causes the lines of latitude and longitude to form a rectangular grid. In such a projection, the ratio between the final x coordinate and the longitude is a constant. This constant is $a*a_{map}$ using the notation from the original question. Thus, changes in scale do not depend on the longitude, so longitude does not appear in the formulas.
It's also a conformal projection, meaning that the distance distortion values for the two dimensions are the same.
More intuitive formulas calculate x and Y directly. For the spherical case, in the equatorial aspect:
$$x=R*a_{map}*(\lambda-\lambda_0)$$
$$y=R*a_{map}*ln \left[ tan \left(\frac{\pi}{4} +\frac{\phi}{2}\right)\right]$$
The ellipsoidal formula is more complicated:
$$y=a*a_{map}* ln \left[ tan \left(\frac{\pi}{4} +\frac{\phi}{2}\right) * \left(\frac{1-e* sin\phi}{1+e*sin \phi}\right)^\frac{e}{2} \right]$$
(It's one of the simplest ellipsoidal projection formulas, however!)
These formulas appear in the Wikipedia article on the Mercator projection and are the ones you should pay attention to.