I'll record here another useful article that is relevant to the topic:$\tag*{}$ Søren Kjærgaard Boldsen, Different versions of mapping class groups of surfaces arxiv.org/abs/0908.2221

@Carl-FredrikNybergBrodda thank you for this notice, I meant "any one". Corrected now.

There exist very short presentations for $B(2,4)$, like the one with 9 relators, see my answer below: mathoverflow.net/a/445712/2164

Why do you think that the element $g$ given as the matrix in the original post, belongs to $GL(V)$ and not to $SL(V)$? It seems that the determinant of $g$ is 1, being the product of diagonal entries.

The above link to Yu's article is not working anymore, but there is one on the Internet Archive: web.archive.org/web/20210506181845/https://…

@YemonChoi Could you tell what a joke about "unstable Adams spectral sequence" is? Google seems to not know it.

I wonder if anyone did the same thing on the hexagonal or triangular plane lattices?

It is true: $PB_3$ is indeed isomorphic to $F_2\times\mathbb Z$. It follows from the fact that $PB_3$ is the pure mapping class group of 3-times punctured disk, and from the fact that the mapping class group of 4-times punctured sphere is $F_2$ (see Farb-Margalit, section 4.2.4). If one takes the 3-times punctured disk and collapses the boundary to a puncture, the mapping class groups surject one another with the kernel $\mathbb Z$, generated by the boundary twist. This short exact sequence splits, as is shown in section 9.3 of Farb-Margalit, so we get $PB_3=\mathbb Z\times F_2$.