Why can we drop the Omega term in Euler's expression of AC signal?
I think you should give the book or docs where the equations come. Maybe it's just for convenient. Because the results of signals addition/subtraction with same frequency also have the same frequency with themselves. 

Below is another version, and i think this version is more clear.

In Fundamentals of Electric Circuits, 5th by Sdiku, Matthew, Page. 378

$$
v(t)=V_{m}cos(\omega t+\phi)=Re(V_{m}e^{j(\omega t + \phi)})
$$

Thus

$$
v(t)= Re(\dot{V}e^{j \omega t})
$$

where

$$
\dot{V}=V_{m}e^{j\phi}=V_{m} \angle \phi
$$

\$\dot{V}\$ is thus the phasor representation of the sinusoid \$v(t)\$.

Note: for clarity, i put one dot on top of \$V\$.



In your equations, apparently, the second and the third equation can't be equal, in your second equation, \$V_{p}\$ should be a scalar, and in your third equation \$V_{p}\$ should be a vector.