Definition: The median of a random variable $X$ is any real value $m$ s.t. $P(X \geq m)\geq\frac{1}{2}$ and $P(X \leq m)\geq \frac{1}{2}$.
This is a strong statement.
How do we prove this?
I always thought this statement is too strong and only believed that a median was a real value $m$ s.t. either $P(X \leq m) =1/2$ and $P(X>m)=\frac{1}{2}$ or it could be true that $P(X \geq m) =1/2$ and $P(X \leq m)=\frac{1}{2}$.
How do we prove the "$\geq \frac{1}{2}$"? part of the definition, which is stronger than just being equal to one-half?
First and foremost, the direct answer to your question is that definitions are not proven, they are stated (as has been commented by two other people), so there is nothing to prove here about the definition of median.
Now, what you are really asking is why do we use this as the definition for median. While I do not personally know, I can prove that both of the definitions you provided are not equivalent characterizations of the median.
Let $X$ be a random variable that takes one of the values in $\{1,2,3\}$, each with equal probability. Then, by definition of a median, we can see that $2$ is the median of $X$, because $\mathbb{P}(X \ge 2) = 2/3 \ge 1/2$, and $\mathbb{P}(X \le 2) = 2/3 \ge 1/2$. So we know that $2$ is a median of $X$ by definition.
Now, in order for any alternate characterization for the median to be true, we must be able to show that $2$ is a median of $X$ in all cases at the very least. Any characterizations for the median that do not result in a median of $X$ being $2$ are definitely wrong, and any which do are only potentially correct.
Let's assume that $m$ is a median of $X$ when $\mathbb{P}(X \le m) = 1/2$ and $\mathbb{P}(X > m) = 1/2$. Since we know that $2$ is a median of $X$, then both of the equations must hold for $m = 2$. However, we can see that $\mathbb{P}(X \le m) = 2/3 \ne 1/2$, so this characterization for the median would imply that $2$ is not a median of $X$, but this contradicts the proven fact that it is, so this characterization can not be valid.
The same goes with the other characterization you provided.
Edit: Changed wording to say "a median" instead of "the median", since technically, there may be several medians in the case that the median is not a value that $X$ can take, though the argument still holds.