Regular categories may equivalently defined as those with:
or
When carefully proving the equivalence, the only limits required are pullbacks i.e. in a category with pullbacks:
coequalizers of kernel pairs & stable regular epis $\iff$ stable regular epi/mono factorization.
Is there a compelling reason to require all finite limits?
A category with pullbacks and equalizers that satisfies the rest of the definition of a regular category is called locally regular, since this is equivalent to saying that all of its slice categories (which of course have terminal objects) are regular in the usual sense. Locally regular categories share many other properties of regular ones, for instance one can construct a bicategory of relations and show that locally regular categories are essentially the same as "tabular allegories" (A3.2.7 in Sketches of an elephant).
There are many reasons one might give for why the notion of "regular category" includes a terminal object (and hence all finite products), but I think one fairly compelling one is that, as David said in a comment, one wants the internal logic of a regular category to be regular logic, and one needs all finite products in order to define a type theory and internal logic: a term $x:A, y:B \vdash t:C$ is a morphism $A\times B\to C$, and a term $\cdot \vdash t:C$ is a morphism $1\to C$. (One can make do with a cartesian multicategory instead, but a locally regular category doesn't have an underlying one of those either.)
