Equational theories determined by "identities without variables" How to characterize equational theories $T$ which have the following property: for any two terms $t(x_1,...,x_n)$ and $t'(x_1,...,x_n)$ in the signature of $T$, if for any closed terms (i. e. terms without variables) $c_1$, ..., $c_n$ the identity $t(c_1,...,c_n)=t'(c_1,...,c_n)$ follows from (the identities of) $T$ then so does $t(x_1,...,x_n)=t'(x_1,...,x_n)$.
In algebraic terms this means to characterize those varieties of algebras which are generated by their initial algebra (the free algebra on the empty set).
The only widely known example of this that I was able to come up with is the theory/variety of Boolean algebras. But in fact any algebra $A$ in any signature generates its own variety of this kind, by adding to the signature a bunch of constants in the well known way and then generating the subvariety by $A$ itself.
So I am interested in any "intrinsic" (say, category-theoretic) characterization of  theories/varieties with the above property, as well as in any other familiar examples of such.
May one hope to actually classify such things up to, say, categorical equivalence?
 A: 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.
A: Aren't these the same as clones containing the constant functions ?
From a clone, you just take the equational theory consisting of all the functions in it, with equations all the relations satisfied by those functions.
From a theory/variety, just take all the operations restricted to the free algebra on the empty set in that varietey.
There are many nontrivial clones on a two-element set, but if I read the diagram right there are only finitely many containing the constant symbols. Among these are:
The clone of monotone functions gives you the variety of bounded lattices.
The clone of affine functions gives you the variety of $\mathbb F_2$-vector spaces with a marked point.
The clone of conjunctive or disjunctive functions gives you the variety of bounded partial orders with joins or with meets.
The clone of unary functions gives you the variety of sets with an involution and two marked elements that are switched by the involution.
The clone of all Boolean operations gives you the variety of Boolean algebras.
The clone of constant functions gives you the variety of sets with two marked elements.
