  A famous result due to Lov\'{a}sz states that two finite relational
structures $M$ and $N$ are isomorphic if, and only if, for all finite
relational structures $T$, the number of homomorphisms from $T$ to $M$ is equal
to the number of homomorphisms from $T$ to $N$. Since first-order logic (FOL)
can describe finite structures up to isomorphism, this can be interpreted as a
characterization of FOL-equivalence via homomorphism-count indistinguishability
with respect to the class of finite structures. We identify classes of labeled
transition systems (LTSs) such that homomorphism-count indistinguishability
with respect to these classes, where "counting" is done within an appropriate
semiring structure, captures equivalence with respect to positive-existential
modal logic, graded modal logic, and hybrid logic, as well as the extensions of
these logics with either backward or global modalities. Our positive results
apply not only to finite structures, but also to certain well-behaved infinite
structures. We also show that equivalence with respect to positive modal logic
and equivalence with respect to the basic modal language are not captured by
homomorphism-count indistinguishability with respect to any class of LTSs,
regardless of which semiring is used for counting.
