In a larger sense, anyone engaged in the examination and theorization of grammatical principles is a student of grammar. Specifically, however, grammar students refer to those individuals whose goal in studying grammar is to understand its rules for the purpose of applying them to writing. Grammar students must receive the teacher-provided definitions, understand how they apply to written and spoken language, and then attempt to reiterate these definitions in their communication. This process of reiteration requires grammar students to redefine grammatical concepts in their own terms so as to develop easy strategies for deploying these concepts as they write or speak. Students distill the definitions of teachers and authors for themselves.
Scaliger's term may refer specifically to teachers at any level tasked with instructing others on the proper use of specific language's grammar. Grammar teachers must not only comprehend what specific grammatical components of a sentence are, but they must also be able to define and explain these components to students. While grammar teachers do not have to develop initial definitions for grammatical components, such as comma splices, run-on sentences, dangling modifiers and so on, they do have to develop working, understandable definitions their students can use. Teachers distill the definitions of grammarians and authors for students.
"Grammar" refers to both the rules that govern written and spoken communication, as well as to the books written that describe these rules. Authors of various grammars -- for example, Funk & Wagnall's "Standard Dictionary of the English Language" -- must articulate and describe grammatical concepts within the pages of their book so that teachers and students can use that book to develop an understanding of grammatical concepts. Authors distill the definitions of grammarians for teachers and students.
Grammar scholars or grammarians offer technical and philosophical definitions of grammatical concepts as they appear in both everyday and formal use. Grammarians develop their definitions of grammatical concepts by searching the structures and uses of language for usage patterns. Once they have identified patterns, grammarians often develop a conceptual frame or term to describe the pattern. Grammarians distill the natural rules that govern grammatical communication for non-grammarians.