2015-09-24
Summarizing, it seems that Heegaard splittings of $0$-manifolds are problematic, or need a different definition, and similarly trisections of $1$-manifolds are problematic; in both cases, the problem is having part of the dimension be $(-1)$-dimensional. But Heegaard splittings of (closed) $1$-manifolds were fine.
Now let's do Heegaard splittings of closed surfaces. Here are two options:
Option 1: A "genus $g$" Heegaard splitting of a closed surface $\Sigma$ is a decomposition $\Sigma=\Sigma_1 \cup \Sigma_2$ such that (1) each $\Sigma_i \cong \natural^g S^1 \times B^1$ and (2) $\Sigma_1 \cap \Sigma_2 \cong \#^g S^1 \times S^0$.
On other words, each $\Sigma_i$ is a planar surface with $g+1$ boundary componets, a.k.a. a $g$-punctured disk (well, disk with $g$ open disks removed). So just put your surface flat on the table and slice parallel to the table, like this:

Note that this doesn't give us much choice. There is only one way to do this for a fixed surface of (actual) genus $h$. And maybe we really want to think of $\Sigma$ as just the double of some surface with boundary, i.e. a $2$-dimensional $1$-handlebody. So...
Option 2: A Heegaard splitting of a closed surface $\Sigma$ is a decomposition $\Sigma=\Sigma_1 \cup \Sigma_2$ such that (1) each $\Sigma_i$ is a $2$-dimensional $1$-handlebody and (2) $\Sigma_1 \cap \Sigma_2$ is a disjoint union of circles.
Now the formalism is breaking down, but this again is something special in low dimensions, that $2$-dimensional $1$-handlebodies are not completely determined by the number of $1$-handles. This suggests also that breaking the formalism a little might help with the $0$- and $1$-dimensional examples that failed. I'll return to this tomorrow, I think. Let's just end with a picture of this kind of Heegaard splitting:

Here each half is a once-punctured torus. All I actually did was cut along the central "neck" first, but then modify that cut by a Dehn twist to make it look fancy.