The general question: how is the boundary defined in product spaces? Given two topological spaces $X,Y$, I'd say that $\partial(X\times Y)=\partial X\times\partial Y$. But looking at what follows it doesn't seem to be right:
Clearly here $\partial_0 P=\partial\Delta_1\times\cdots\times\partial\Delta_n$, but it doesn't coincide with $\partial P$.
So, what's the difference between $\partial P$ and $\partial_0 P$?
You can see that that $\partial(X\times Y) \neq \partial X\times\partial Y$ by considering $X = Y = (0, 1) \subset \mathbb{R}$. On the one hand, $\partial(X\times Y)$ is the boundary of the unit square, while $\partial X\times \partial Y$ consists of the four corner points.
The relationship between the boundary of a product and the boundaries of the factors is
$$\partial(X\times Y) = (\partial X\times\overline{Y}) \cup (\overline{X}\times\partial Y).$$
For the example above, $\partial X = \partial Y = \{0, 1\}$ and $\overline{X} = \overline{Y} = [0, 1]$, so
$$\partial(X\times Y) = (\{0, 1\}\times[0, 1])\cup([0, 1]\times\{0,1\})$$
which indeed is the boundary of the unit square: $\{0, 1\}\times[0, 1]$ gives the two vertical sides, and $[0, 1]\times\{0, 1\}$ gives the two horizontal sides.
Note that in this example, $\partial X\times \partial Y \subseteq \partial(X\times Y)$. This is also true in general, as can be seen from the formula above; in fact, $\partial X\times\partial Y = (\partial X\times\overline{Y})\cap(\overline{X}\times\partial Y)$.