There are some difficulties with understanding the various words in the group linked with linguistic.
First, its use in general English in academic circles is as an adjective, with the related adverb linguistically. Here, it means 'to do with language and how we talk about it.' A lecturer may say to a student "Your essay was excellent in its content and argument, but linguistically it was very poor." This means that it was a good essay, apart from the way the ideas were expressed in language. The most widely used meaning of 'a linguist' is 'a person who speaks a second or other language', with the implied addition 'well'. In academic circles, students whoa re studying languages are often called linguists.
This broad meaning has a more carefully refined meaning. A linguist, can be "a student [or teacher, or academic generally] of linguistics". Linguistics is the study of language in a broader sense: it is about languages in general, not of any one language on particular. It considers theory and the 'universals', or features held in common by all human languages. A linguist of this sort studies grammar, etymology and other theoretical aspects of linguistics. (See also linguistics.) It may be more advisable to call such people linguisticians, to avoid ambiguity - although it has been said that the word "implies pretentiousness rather than precision": cited in OED, s.v. linguistician, anonymous quotation 1950.
Like the words Mathematics and Economics, linguistics is the name of a subject which looks like a plural, although it is usually treated as a singular grammatically.