I know that the original publication ' Alfred Horn (1951), "On sentences which are true of direct unions of algebras" ' didn't require universal quantification. However, it didn't call these Horn clauses either.
My question arises, because "Mathematical Logic" from Ebbinghaus et al. omits universal quantification for Horn clauses. Everybody else seems to use Horn clauses as if they were always universally quantified. (Even Ebbinghaus et al. only treat universal Horn clauses, except for an exercise that asks to prove a result from the original publication.)
Now I wonder: Are Horn clauses always universally quantified, or do I have to explicitly talk about universal Horn clauses?
Edit The statements made in this question are wrong. I read the book in German and got the meaning of "clause" wrong. I decided against deleting the question or trying to correct it by editing. Instead, I wrote an answer which tries to explain the confusion.
The cited book doesn't even mention Horn clauses. It only talks about (predicative) Horn formulas and Horn sentences. Its definition of (predicative) Horn formulas agrees well with common practice. A Horn formula is a conjunction of Horn clauses. Sadly, "universal Horn sentence" is sometimes used synonymously to "(universal) Horn clause".
In conclusion, the question was just the result of a confusion between "clause" and "formula". This confusion was partly also caused by reading the book in German, and didn't paying attention to "silent" redirections by google and wikipedia.
There is actually both a connection between the closure under products of Horn structures and universal quantification. The closure under products doesn't require universal quantification, but the closure under products suggests a canonical way to define homomorphisms: a function is a homomorphism, if its graph also satisfies the axioms of the structure. The graph is a subset of the product. Without universal quantification, this sort of homomorphisms wouldn't be closed under composition.