What is the standard form of a linear programming (LP) problem?

I have seen both the $\min$ and $\max$ forms of an LP frequently, it seems to be an author preference sort of thing. The only difference is a minus sign in the objective ($-c^Tx$ instead of $c^Tx$).

Regarding the constraints, I have more often seen the first form (your Bertsimas reference) referred to as standard or canonical. The two forms are equivalent in some sense.

Since $Ax=b$ can be written as the pair of inequality constraints $Ax \leq b$ and $(-A)x \leq (-b)$, it is clear that the first form can be expressed directly as a problem of the second form.

The inequality $Ax\leq b$ can be written as a combination of an equality $Ax+ \sigma = b$ and an inequality $\sigma \geq 0$. Hence by increasing the number of variables (ie, using the variables $x$ and $\sigma$), we can express the second form as a problem of the first form, ie, $\begin{bmatrix} A & I \end{bmatrix} \pmatrix{x \\ \sigma} = b$, $\pmatrix{x \\ \sigma} \geq 0$.

The problem $\min \{ c^T x | A x \leq b \}$ is sometimes referred to as an inequality form LP. Again, it is equivalent to the other two forms.