I don't know the answer to this question, but
I have a correction and a remark.

The question asks "When is a variety $\mathcal V$ generated by
its free algebra over the empty set?"
It has been asserted that this question is equivalent
to the question of when ${\mathbf F}_{\mathcal V}(\emptyset)$ is
existentially closed in $\mathcal V$.

**The correction.**
The assertion is not true. There is an implication, but not
an equivalence.

Let ${\mathbf F}_0:={\mathbf F}_{\mathcal V}(\emptyset)$,
and assume that $\mathbf F_0$ is existentially closed (e.c.) in $\mathcal V$.
$\mathbf F_0$ is embeddable is $\mathbf F_m:=\mathbf F_{\mathcal V}(m)$ for every $m$.
If the q.f. formula $s(\bar{x})\neq t(\bar{x})$ is satisfiable in $\mathbf F_m$,
then it is satisfiable in $\mathbf F_0$ by the e.c. property.
This shows that $\mathbf F_0$ fails every identity that fails
in $\mathcal V$, while the fact that $\mathbf F_0\in{\mathcal V}$
shows that it satisfies every identity
that holds in $\mathcal V$. Altogether this shows that
$\mathbf F_0$ e.c. implies $\mathcal V$ is generated by $\mathbf F_0$.

But the converse is false. The variety of commutative rings is
generated by ${\mathbf F}_0 = \mathbb Z$,
but this ring is not e.c. in the variety of commutative rings.
Moreover, there are many nontrivial varieties generated by their
initial algebras, $\mathbf F_0$, where these initial algebras
happen to be finite.
(E.g., the variety of Boolean algebras, or bounded distributive lattices,
or the variety generated by the ring of integers modulo $n$, or
any variety generated
by the constant expansion of a finite algebra.)
But a nontrivial finite algebra cannot be e.c.

**The remark.**
The original poster asks if there is a category-theoretic
characterization of these varieties. Well, there is one,
since you can express Birkhoff's HSP Theorem category-theoretically.
To express $\mathcal V = \mathbf{HSP}(\mathbf F_0)$ you just need
to say that every object in $\mathcal V$
is the image of an extremal epimorphism
from some object that has a monomorphism into some power
of the initial algebra.
But, I doubt that there is a nontrivial characterization of these varieties.
As the original poster noted, any variety generated by the constant
expansion of an algebra has the desired property, so this class
of varieties
is as varied as the class of constant expansions of algebras.