$$ \langle a,b,c\mid a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=1\rangle =\Delta(p,q,r) $$ This is a presentation of a triangle group $\Delta(p,q,r)$, a special kind of Coxeter group.
EDIT In fact, these are called extended triangle groups, by G. Jones and D. Singerman in Maps, hypermaps and triangle groups...
What about the following presentation: $$ \langle a,b,c\mid a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=(abc)^s=1\rangle $$ Do these groups have a name and where are they treated?
The presentation in question are motivated by this and that...
ANOTHER EDIT if $p=q=r$ is prime and $s=1$ this is called triangular Fuchsian group here...
I haven't come across a name for this family in full generality, but the special case in which $p=2$ was defined and studied by Coxeter in his paper
H. S. M. Coxeter, The abstract groups $G^{ m, n, p}$, Trans. Amer. Math. Soc. 45 (1939), 73-150.
where (in your notation) the group is called $G^{q,r,s}$.
Also, when $s$ is even, your group has a subgroup of index $2$ with presentation $\langle x,y \mid x^p=y^q=(xy)^r=[x,y]^{s/2} \rangle$.
These groups were studied in the same paper by Coxeter, and denoted $(p,q,r;s/2)$.
Both of these families have been extensively studied since then, in particular concerning their finiteness. They are generally infinite for sufficiently large values of the parameters, and there is just a handful of remaining cases for which their finiteness is still unknown.
A few years ago Havas and I showed, using a big computer calculation, that $(2,3,13;4)$ is finite of order $358\,848\,921\,600$. So your group with $(p,q,r,s) = (2,3,13,8)$ has twice that order.