Relationship Between Sampled Continuous and Discrete Time Signals
You've shown that the (continuous-time) Fourier transform (CTFT) of a sampled continuous-time signal equals the discrete-time Fourier transform (DTFT) of the corresponding discrete-time signal. In both cases, the spectrum is periodic. The independent variables are related by

$$\omega=\Omega T\tag{1}$$

where $\Omega$ is the angular frequency ($\Omega=2\pi f$), $T=1/f_s$ is the sampling period ($f_s$ is the sampling frequency), and $\omega$ is the normalized angular frequency, which is often used as the independent frequency variable when describing spectra of discrete-time signals. Note that you could also use $\Omega$ as the independent variable for discrete-time signals, but it always appears together with the sampling interval $T$. This is an important property of sampled or discrete-time signals, and that's also why the two variables $\Omega$ and $T$ are commonly combined into a single variable $\omega=\Omega T=2\pi f/f_s$.

Note that sampling results in a periodization of the spectrum, and this is why information can be lost. This is true for both spectra $X_s(j\Omega)$ and $X(e^{j\omega})$. The periodicity of the spectra implies that the fundamental frequency interval $f\in [-f_s/2,f_s/2]$, or, equivalently, $\Omega\in [-\pi/T,\pi/T]$ or $\omega\in [-\pi,\pi]$ contains all information of the sampled signal, and all frequencies outside that interval are redundant. This is an inherent property of sampled (or  discrete-time) signals, and it is the cause of aliasing, i.e. the mapping of different continuous-time signals to the same sampled signal.