What is the difference between modulating with cosine and exponential function?
Considerations:

Instead of re-writing all the math, I'd like to direct you to this answer by @Matt L. over in math stackexchange to help you understand what happens when you multiply by cosine.
Meaning of multiplying by sin

Notice the 1/2 that shows up when using just cosine.

Answer:


  If we only look at the real part we get almost the same as above with a cosine but it's perfectly symmetrical. What I don't understand is why?


You're right, if you only look at the real part when multiplying by the exponential then you in fact get the exact same thing as cosine (except the magnitude). 

Even in your expansion using Euler's formula you can see this:

$$ e^{jwt} = Re{(e^{jwt})} + Im{(e^{jwt})} $$
$$ e^{jwt} = cos(wt) + jsin(wt) $$
$$Re(e^{jwt}) = cos(wt) $$

You are also correct in your drawing that using only one exponential will create only a single shift in one direction. Furthermore, this single multiplication (as in your second drawing) shifts the full power to right side and hence you don't see a 1/2 magnitude multiplication (like you see with the cosine).

This is where you must pause and realize that this multiplication in practice is actually not possible. We can create phase shifts but you cannot make sin because it is inherently not real, and thus you cannot make the signal in the second drawing because it isn't real. 

So multiply by \$e^{jwt}\$ as much as you'd like but remember when you actually take this to the field you will only see the real quantities.