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) <= 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?