Uniqueness of Fourier Series Representation and the Fourier Transform of Periodic Signals If we are given a signal of the form $$x(t) = \sum_{k = -\infty}^{+\infty} a_k e^{j k \omega_0 t},$$ can we call it a Fourier Series representation of $x(t)$ right away?

Suppose we are given the signal $x(t) = e^{j2\Omega t} + e^{j4\Omega t}$, this signal can be expressed as $$x(t) = \sum_{k = -\infty}^{+\infty} a_k e^{jk\Omega t},$$ however, in this case, $2\Omega$ is the fundamental frequency of the signal, not $\Omega$ as the equation above might suggest.

I ask this because in the second edition of Signals and Systems by Alan Oppenheim, he derives the Fourier transform of a periodic signal by considering the impulse train $$X(j\omega) = \sum_{k = -\infty}^{+\infty} 2\pi a_k\delta (\omega - k \omega_0)$$
and applying the inverse Fourier transform to obtain $$x(t) = \sum_{k = -\infty}^{+\infty} a_k e^{j k \omega_0 t},$$ which he says is the Fourier series representation of the signal without further discussion. But how do we know that in this case $\omega_0$ is the fundamental frequency of the signal? Couldn't it be the case that it is a signal of the form given above, namely, $x(t) = e^{j2\omega_0 t} + e^{j4\omega_0 t}$, in which case the fundamental frequenct is $2\omega_0$? Thank you in advance.
The Fourier series, if it exists, is unique, apart from the fact that we can always use a multiple of the fundamental period $T$ as the period, or, equivalently, a fraction of the fundamental frequency $\omega_0$. The only consequence of this is that we introduce Fourier coefficients with value zero in between the non-zero coefficients. The values of the non-zero coefficients remain unchanged:

$$x(t)=\sum_{k=-\infty}^{\infty}a_ke^{jk\omega_0}=\sum_{k=-\infty}^{\infty}\tilde{a}_ke^{jk\omega_0/N},\quad N\in\mathbb{Z}^+$$

$$\tilde{a}_k=\begin{cases}a_{k/N},\quad k=lN,\quad l\in\mathbb{Z}\\0,\quad\text{otherwise}\end{cases}$$