Power Spectrum: Definition
The first definition of the power spectrum is the correct one (at least for causal signals, otherwise you need symmetrical integration limits). The second one assumes that the Fourier transform of $x(t)$ exists, which is generally not the case for random signals. The fact that the power spectrum and the autocorrelation function form a Fourier transform pair is no contradiction with the first definition of the power spectrum. This can be shown as follows:

$$S_x(\omega)=\lim_{T\rightarrow\infty}E\left\{ \frac{1}{T}\left| \int_{-T/2}^{T/2}x(t)e^{-j\omega t}dt \right|^2 \right\}=\\=
\lim_{T\rightarrow\infty}E\left\{ \frac{1}{T} \int_{-T/2}^{T/2}x^*(t)e^{j\omega t}dt \int_{-T/2}^{T/2}x(t')e^{-j\omega t'}dt'  \right\}$$

The inverse Fourier transform of the power spectrum $S_x(\omega)$ should equal the autocorrelation function $R_x(\tau)$:

$$\mathcal{F}^{-1}\{S_x(\omega)\}=\frac{1}{2\pi}\int_{-\infty}^{\infty}S_x(\omega)e^{j\omega \tau}d\omega=\\=
\lim_{T\rightarrow\infty}E\left\{ \frac{1}{2\pi T} \int_{-\infty}^{\infty}\int_{-T/2}^{T/2}x^*(t)e^{j\omega t}dt \int_{-T/2}^{T/2}x(t')e^{-j\omega t'}dt'e^{j\omega \tau}d\omega  \right\}=\\=
\lim_{T\rightarrow\infty}E\left\{ \frac{1}{T} \int_{-T/2}^{T/2}x^*(t) \int_{-T/2}^{T/2}x(t')\left[\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{j\omega (t+ \tau-t')}d\omega\right]dtdt'  \right\}
$$
The expression in brackets equals $\delta(t+\tau-t')$, which gives

$$\mathcal{F}^{-1}\{S_x(\omega)\}=
\lim_{T\rightarrow\infty}E\left\{ \frac{1}{T} \int_{-T/2}^{T/2}x^*(t) x(t+\tau)dt  \right\}=\\
=\lim_{T\rightarrow\infty} \frac{1}{T} \int_{-T/2}^{T/2}E\{x^*(t) x(t+\tau)\}dt=\\
= \lim_{T\rightarrow\infty} \frac{1}{T} \int_{-T/2}^{T/2}R_x(\tau)dt=
R_x(\tau)\lim_{T\rightarrow\infty} \frac{1}{T} \int_{-T/2}^{T/2}dt=R_x(\tau)
$$

which shows that the chosen definition of the power spectrum is compatible with the fact that the power spectrum and the autocorrelation function are Fourier transform pairs. A similar derivation can be done for discrete-time signals.