One of the eternal conflicts in linguistics is descriptivism versus prescriptivism. The two are different approaches to the study of language. Prescriptivism asserts that in order to be useful as a mediator of social discourse, language must have rules. Those rules must be commonly agreed upon, and all parties must abide them or communication is made more difficult and faces a greater probability of failure. Descriptivism asserts that linguistics is the study of languages as they are used in the real world by humans. That is, the native speakers of a language are the ultimate arbiters of the "rules" of a language, and that if a given language communication succeeds, then the language must perforce have been used correctly. The two approaches are somewhat in conflict, and most reputable linguistic schools chart a middle course involving elements of each approach.
Word meaning is as amenable as any other sub-field to both approaches. Prescriptivism suggests that words mean what they mean, and if a given word is used to mean something else, then that word is used incorrectly. Meanwhile descriptivism would suggest that if meaning is successfully conveyed, then the word was used correctly.
Following Tarski, a formal approach to lexical semantics sees words as defining sets. Of the universe of discourse, the word defines a set: all those phenomena that are meant by the word are in that word's set, all phenomena not meant by that word are in that sets complement. Or, to borrow vocabulary from semiotics, all those denotata referenced by a sign are in that sign's set, while all other denota are in that set's complement. For example, if the word 'cat' is defined as all filidae with four legs, fuzzy ears, and a desire for ear scritchies, then all phenomena (denotata) that fulfill that definition are in the set of all cats.
Sets, of course, have two different definitions. The first is the intensional definition, which lists the predicates all members of the sat share. The second is the extensional definition, listing all members of the set. Should those two defenitions ever fail to correspond, one or both should be modified (we are ignoring, for the moment, any theories on fuzzy set membership).
Of interest when defining words is identifying edge cases, which may or may not be members of the set. Whether a native speaker of the language would identify a given edge phenomena as belonging to the set or not shapes the defenition of the word that the set represents. A difficulty arises when two or more native speakers disagree on set membership for a given phenomenon, this can lead to communicative difficulties and even to controversy (is a human zygote included in the set of all persons? Why or why not?). Language communication succeeds most often where edge cases are avoided.
Language changes over time. One of the processes of that change is the alteration of word meaning. Word meaning changes as some phenomena that were not previously included in membership in a set are now included, or when some phenomena previously inclded within a set are judged to no longer be included (or whan their fuzzy inclusion values are changed).