Is this the correct "standard form" of nonlinear programming (optimization) problem and if it is why it's in this form? Rather a simple question I guess, though makes me wonder. The standard form I've found in the book (and on wiki) is something like this:

$\min f(x)$

s.t.

$h_i(x) = 0$

$g_i(x) \le 0$

Is this considered a "standard form" for nonlinear optimization problems? And if it is why it's defined like this? Why it has to be exactly the min of the function and why constraints have to be either equal or less than 0 or equal to 0? I couldn't find any answer why it is as it is actually. Is there some important thing why it couldn't be max actually for example?
There is no "standard form" that you must use. You don't have to write an optimization problem in that way. This is a useful representation in a number of ways; it is very general; and it facilitates some kinds of analysis (e.g., KKT conditions).  But there are no police forcing you to express your optimization problem in this way; you can use whatever you find most useful.

Note that every optimization problem can be expressed in this way, as $h(x)=0$ can express arbitrary constraints on $x$ and $f(x)$ can express an objective function.  So, it is nice from this perspective.