I'm trying to quickly learn basic crystallographic principles in 2D under pressure (it's me whose under pressure, not the lattice) so I'm checking many sources to find those that "speak to me" the best (including this excellent answer). Many sources (books, papers, researcher's websites) cite the "honeycomb lattice" as an example of a lattice which is non-Bravais in order to reinforce their explanation of what a Bravais lattice is.
ambiguous or noncommittal as to being a "lattice" slide 6
Briefly (I don't have the various texts with me right now to transcribe the paragraphs) it seems to me that a 2D Bravais lattice any set of points $m \vec a + n \vec b$ for all integers $m, n$.
The vertices of a honeycomb don't satisfy this because if you try to use that expression, 1/3 of the points are missing. If you want to use a Bravais lattice, you have to define a unit cell with two "atoms" in it.
Question: Can the vertices of a honeycomb or "honeycomb lattice" really be called a proper lattice (but just not a Bravais lattice), or should we really call it either a net† or a regular complex apeirogons or something else, but not refer to it as a lattice?
From http://www-personal.umich.edu/~sunkai/teaching/Winter_2018/01302018.pdf (click for full size)
†The Five nets
In triperiodic structures the equivalent points form a three-dimensional lattice in which the space units are unit cells. In diperiodic structures the equivalent points form a two-dimensional net in which the area units are unit meshes. The fourteen Bravais space lattices are replaced, in the diperiodic case, by five nets, described in Table I and diagrammed in Fig. 2.