I'm reviewing some vector calculus material in Chapter 16 of Stewart's Calculus (9th ed) and I want to make sure I understand what is meant by a surface being oriented in the direction of the positive (or negative) $y$ (or $x$ or $z$) axis. Here are some examples from section 16.8 of Stewart:
S is the part of the hemisphere $x^2 + y^2 + z^2 = 4, \,y \geq 0$ oriented in the direction of the positive $y$-axis
$S$ is the cone $x = \sqrt{y^2 + z^2}, \, 0 \leq x \leq 2$ oriented in the direction of the positive $x$-axis.
Does "oriented in the direction of the positive $y$-axis" mean that our choice of unit normal vector $\mathbf{n} = \langle n_x, n_y, n_z \rangle$ is such that $n_y > 0$? If so, then I assume that orientation in the direction of the $-y, \pm x$ and $\pm z$ axes are defined similarly?
Also, am I correct in thinking that this terminology for orientation only applies in the case where $S$ is given by the graph of some bivariate function? For example, "oriented in the direction of the $+y$-axis" only makes sense for a surface $S$ of the form $S = \{(x,y,z) \in \mathbb{R}^3 : y = h(x,z), (x,z) \in D\}$. Is this correct?