I should explain how to think about these things in terms of Frobenius splitting. This perspective will let you write down lots more examples like the ones you already have. For example, if I haven't made any silly typos, $x - 2 + 3x^{-1} - x^{-1} y - x^{-1} y^{-1}$ should be an example.

$\def\Tr{\mathrm{Tr}}$
Here is the strategy. Let $\Tr$ be the following map from Laurent polynomials to themselves given by
$$\Tr\left( \sum g_{i_1 i_2 \cdots i_n} x_1^{i_1} \cdots x_n^{i_n} \right) = \sum \sum g^{1/p}_{(p i_1) (p i_2) \cdots (p i_n)} x_1^{i_1} \cdots x_n^{i_n}.$$
Given any Laurent polynomial $f$, we define an almost splitting by $\phi_f(g) =\Tr(f^{p-1} g)$. This is a splitting if and only if $\Tr(f^{p-1}) =1$, meaning that the constant term of $f^{p-1}$ is $1$ and there are no other monomials in $f^{p-1}$ whose exponents are divisble by $p$.

Consider the following:

**Condition 1:** Let $N(f)$ be the Newton polytope of $f$, i.e., the convex hull of the monomials occurring in $f$. Then $(0,0,\ldots, 0)$ is the only lattice point in the interior of $f$.

In the presence of Condition 1, the only monomial in $f^{p-1}$ whose exponents are all divisible by $p$ is the constant term. So, in the presence of Condition 1, what you want to know is whether $\phi_f$ is a splitting. From now on, we assume Condition 1.

Now, there is a very powerful tool to force $\phi_f$ to be a splitting. If there is an $\mathbb{F}_p$ point $z$ of $f=0$, where the hypersurface $f=0$ has normal crossings, then $\phi_f$ will be a splitting. (This is in, for example, Brion and Kumar's book.) We can also weaken this condition in various ways. It is enough to have residual normal crossings (see V. Lakshmibai, V. B. Mehta, and A. J. Parameswaran, Frobenius splittings and blow-ups, J. Algebra 208 (1998), no. 1, 101–128. MR 1643983 (99i:14059)). Even more generally than this, let $y_1$, ..., $y_n$ be local coordinates at $z$; it is enough that there is some term order (with $1< y_1$, ..., $y_n$) such that $y_1 y_2 \cdots y_n$ is the minimal term of $f$ near $z$. Moreover, if the point $z$ only exists over an extension $\mathbb{F}_q$, or if the coordinates in which $f=0$ becomes normal crossings are only defined over $\mathbb{F}_q$, there are modifications of these results available, although I don't know quite what they say off the top of my head.

I realize that I am not including as many details as I perhaps should here, but it is a large topic. If you give me some clues as to how far you've gotten, I can adjust accordingly.

Now, I want to convince you that all your examples where the constant term is inside the Newton polygon are of this sort. Namely: $x-x^{-1}$ has two roots, $1$ and $-1$, both defined over $\mathbb{F}_p$ for any odd $p$. So $f=0$, near either of these points, is a zero dimensional "normal crossings" variety. If you instead work with $x+x^{-1}$, then the roots are only defined over $\mathbb{F}_p$ when $p \equiv 1 \mod 4$, which is why you got peculiar behavior depending on what $p$ was mod $4$.

Your third example factors as $(x_1-x_2)(1-x_1^{-1} x_2^{-1})$, the intersection of a line and a hyperbola. This has simple normal crossings at $(1,1)$ and $(-1, -1)$.

ARupinski's example is $n$ hypersurfaces which all meet transversely at $(-1,1,-1,1,\cdots,-1,1)$, if I didn't screw up any signs.

The example I gave at the beginning is a nodal cubic, with node at $(1,1)$, if I didn't screw it up.

What would be interesting is if anyone knew an example of this behavior which *wasn't* explained by a singularity of the hypersurface.