So far, when I was learning surface code the patches on which my qubits were had a "rectangular" shape and they involved stabilizers having either $3$ or $4$ "legs". I was then able to easily identify the smooth or rough boundaries given the definition provided for instance in this paper.
A smooth boundary is a boundary where the data qubits belong are "inside" an $X$-stabilizer that has 3 terms, and the $Z$-stabilizer has $4$ terms.
A rough boundary is a boundary where the data qubits belong are "inside" an $X$-stabilizer that has 4 terms, and the $Z$-stabilizer has $3$ terms.
For instance on the surface represented below (a $Z$ stabilizer is at the centre of the plaquette, an $X$ stabilizer is at the node, data qubits are at the middle of the edges), the top/bottom boundaries are smooth and the left/right are rough.
My confusion starts when I switch to another way to represent surface code (I am much less familiar with it and I get easily confused) ( * ). I consider the way to represent patches as described in this paper. In the appendix, we have the following graph that provides the color-convention to define $X$ and $Z$ stabilizers. The stabilizer qubits are at the middle of the associated colored surface. The data qubits are at the node of the lattice.
Here, we cannot apply the previous definition as now some stabilizers have only two "legs". In this case I am a bit confused to know how the smooth/rough notions are defined (and in all papers I have read they seem to assume it is a known definition).
Is it defined from the construction of the logical $X$ and $Z$ operator on the boundaries? Ie, a smooth boundary is defined by a boundary where I can define my $X$ operator. A rough boundary is defined by a boundary where I can define my $Z$ operator. Is there a paper where there is a simple definition provided?
( * ) Technically speaking this is just a rotation of angle $\pi/4$ of the first representation where we additionally remove some qubits, see for instance this post to see the connection between the representations.
Comments regarding Craig Gidney's answer
If I understood correctly Craig's slides, the $X$ and $Z$ boundaries would correspond as the red and purple data qubits shown below. We can also notice that some qubits will belong to both an $X$ and $Z$ boundary.
What confuses me is that in this paper, he doesn't do the same identification (see the image at the very bottom).
Does that mean that there are simply different definitions going on in the literature and I shouldn't pay too much attention to the terminology?
A more simple explanation would be that I basically misunderstood the answer =)