An auxiliary is a non-°affixed °bound form cooccurring with a °verb that expresses TAMEP (°tense, °aspect, modality, °evidentiality, or polarity) meanings and that has person marking which °indexes the verb’s °subject.
The definition is from Haspelmath (2027). It does not define "auxiliary" as a kind of verb, but it requires the presence of subject indexing in order to distinguish auxiliaries from verb-associated particles. It does not make reference to a continuum or to grammaticalization, but defines an auxiliary as a bound form that is not an affix (i.e. a clitic).
auxiliary (CXN) = the element expressing TAMP meaning in an auxiliary construction. [Example: in The cats have eaten, have is the auxiliary in the auxiliary construction have eaten.] (Section 13.4) ❡
auxiliary construction (CXN) = an eventive complex predicate construction in which one element of the construction, the auxiliary, denotes tense, aspect, modality, and/ or polarity (typically abbreviated TAMP), and the other element of the construction denotes the event whose tense, aspect, modality, and/or polarity is expressed by the first element. [Example: in English She might be sitting in the living room, might be sitting is an example of an auxiliary construction.] (Section 13.4)