After looking around more on Wikipedia, the answer appears to be:
- If the same sequence of graphemes can serve as different parts of speech (e.g. the noun 'run' vs. the verb 'run'), those are separate lexemes.
- If, within a single part-of-speech, the sequence of graphemes has multiple etymologically-separate possible meanings, those are separate lexemes. Homonyms are separate lexemes.
- If, within a single part-of-speech, some group of definitions for a grapheme sequence are very closely related in meaning and also etymologically closely related, then those different definitions do not constitute separate lexemes, but are instead considered different "senses" within a single polysemous lexeme.
A lexeme belongs to a particular syntactic category, has a certain meaning (semantic value), and in inflecting languages, has a corresponding inflectional paradigm; that is, a lexeme in many languages will have many different forms.
A syntactic category is a type of syntactic unit that theories of syntax assume. Word classes, largely corresponding to traditional parts of speech (e.g. noun, verb, preposition, etc.), are syntactic categories.
A distinction is sometimes made between true homonyms, which are unrelated in origin, such as skate (glide on ice) and skate (the fish), and polysemous homonyms, or polysemes, which have a shared origin, such as mouth (of a river) and mouth (of an animal). (...)
A further example of a homonym, which is both a homophone and a homograph, is fluke. Fluke can mean:
- A fish, and a flatworm.
- The end parts of an anchor.
- The fins on a whale's tail.
- A stroke of luck.
These meanings represent at least three etymologically separate lexemes, but share the one form, fluke.
A polyseme is a word or phrase with different, but related senses. (...) The difference between homonyms and polysemes is subtle. Lexicographers define polysemes within a single dictionary lemma, numbering different meanings, while homonyms are treated in separate lemmata. Semantic shift can separate a polysemous word into separate homonyms. For example, check as in "bank check" (or Cheque), check in chess, and check meaning "verification" are considered homonyms, while they originated as a single word derived from chess in the 14th century.
In morphology and lexicography, a lemma (plural lemmas or lemmata) is the canonical form, dictionary form, or citation form of a set of words (headword). In English, for example, run, runs, ran and running are forms of the same lexeme, with 'run' as the lemma. 'Lexeme', in this context, refers to the set of all the forms that have the same meaning, and lemma refers to the particular form that is chosen by convention to represent the lexeme.