Carl Linnaeus, who formalized binomial nomenclature, made explicit distinctions between specific, generic, and trivial names.
For example the binomial name of the tiger, Panthera tigris:[1] This was the proper usage from the 18th century into the late 20th century, although many authors seemed to be unaware of the distinctions between trivial and specific names and inconsistent and erroneous usage even appeared in the International Code of Zoölogical Nomenclature.
[1] Grammatically, a binomen (and a trinomen, also) must be treated as if it were a Latin phrase, no matter which language the words were originally taken from.
(This gives some justification to the popular usage of the phrase "Latin name" instead of the more correct phrase "scientific name".)
The specific name must adhere to certain conventions of Latin grammar.