What does 2-flat mean when discussing APN permutations?
In the context of Boolean functions, "flat" is usually used as a synonym for "affine subspace of $\mathbb{F}_2^n$". More generally, an $n$-flat in a vector space $V$ (which may be considered as an affine space) is an $n$-dimensional affine subspace of $V$.

Note that a 2-flat of a vector space $V$ over $\mathbb{F}_2$ is then any set of the form

$$A = \{a, a + b, a + c, a + b + c\}$$

with $a, b, c \in V$ distinct. Remark that $\sum_{x \in A} x = 0$ and conversely, any four distinct elements in $V$ that sum to zero form a 2-flat.