How to prove these two definitions of the minimum phase transfer function are same?
To your second definition it should be added that you only consider causal transfer functions, because it is not difficult to find a smaller phase lag with a non-causal system:


  A minimum-phase system is a causal and stable system with a phase lag that is smaller than the phase lag of any other causal and stable system with the same magnitude response.


Note that a real-valued zero in the left half-plane contributes a phase change of $\pi/2$ to the total phase as we move along the frequency axis from $\omega=0$ to $\omega\rightarrow\infty$:

$$0\le\arg\{j\omega+a\}&lt;\frac{\pi}{2},\qquad a&gt;0,\quad\omega\in[0,\infty)\tag{1}$$

A complex conjugate pair of zeros contribute a phase change of $\pi$.

On the other hand, a real-valued zero in the right half-plane contributes a phase change of $-\pi/2$ as $\omega$ moves from zero to infinity:

$$-\frac{\pi}{2}&lt;\arg\{a-j\omega\}\le 0,\qquad a&gt;0,\quad\omega\in[0,\infty)\tag{2}$$

Note that I've chosen the sign of the term $s\pm a$ such that in both cases the phase is zero for $\omega=0$, which is necessary for a fair comparison between the two cases.

Consequently, exchanging a zero in the left half-plane for a zero in the right half-plane (without changing the magnitude of the frequency response) will always result in an additional phase lag of $\pi$ for $\omega\rightarrow\infty$.

As an example, consider two first-order transfer functions:

$$H_1(s)=\frac{s+2}{s+1}\quad\text{and}\quad H_2(s)=\frac{2-s}{s+1}\tag{3}$$

The signs of $H_1(s)$ and $H_2(s)$ were chosen such that their phases are both zero for $s=0$ (and not $\pm\pi$). The figure below shows the phase plots ($\arg\{H_1(j\omega)\}$ in blue, and $\arg\{H_2(j\omega)\}$ in green):



The pole contributes a phase change of $-\pi/2$ as $\omega$ moves from zero to infinity. The left half-plane zero of $H_1(s)$ contributes a phase change of $\pi/2$, resulting in a net phase change of zero, whereas the right half-plane zero of $H_2(s)$ contributes a phase change of $-\pi/2$, resulting in a total phase change of $-\pi$.

Another way to see the same thing is to note that any causal and stable transfer function can be written as the product of the minimum-phase transfer function with the same magnitude and a causal and stable allpass:

$$H(s)=H_m(s)H_a(s)\tag{4}$$

It can be shown that the phase of a causal and stable allpass is always non-positive for $\omega\in[0,\infty)$, and, consequently, the phase lag of the minimum-phase system is always less than or equal to the phase lag of any other causal and stable system with the same magnitude response.