What is the difference between definition of an algebra on $V$ when $V$ is a $K$-module ($K$ is field) and when $V$ is a vector space?
Let us consider Leibniz algebras:
A Leibniz algebra over $K$ is a vector space $V$ equipped with a binary operation satisfying Leibniz identity.
There exists another definition saying:
A Leibniz algebra over $K$ is a module $V$ equipped with binary operation satisfying Leibniz identity.
What if any is the difference?
There is no differences. For $K$ a field, $K$-module and $K$-vector space mean the exact same thing.
The two definitions you quote are the same; they just use two different words that have the same meaning.
(Maybe one could argue that the second does not include the information that $K$ is a field, while it is implied by the first. But I assume the intent is that $K$ being a field was stated earlier.)
A vector space is a module over a field. Or differently, a module is the generalization of vector space to the case where the scalars do not form a field but a ring.
For the definition of algebra, it is not uncommon to have it over commutative rings, and thus one uses the terminology module. This then might be preserved even when the ring happens to be a field.