The usage difference between [w̥] and [ʍ] is primarily in the phonological status of the segment. Given a choice between representing a phoneme with a bare symbol versus with a symbol plus diacritic, the bare symbol is preferred. If the language in question phonemically contrasts [w̥] and [ʍ] then you have to use both. If there is just one always-voiceless phoneme, which contrasts with [w], it is preferred to use [ʍ]. If there is a single phoneme that has both voiced and voiceless allophones, it is preferred to write [w̥] for the voiceless allophone. In English "twit", devoicing is an allophonic variant of the phoneme /w/, therefore preferred usage is to write [w̥]. This is reinforced by consideration of the full set of facts of devoicing, where there is also devoicing or /r, l, j, m, n/ (e.g. "clay, snore, p[j̥]ewter" – there is no bare symbol for voiceless nasals or r so the diacritic is mandatory).
Some dialects of English have a phoneme /ʍ/, found in which, when, whim. This would support using ʍ for the autonomous phoneme and w̥ for the contextual allophone. Since the two sounds never appear in the same context, you can't establish that the sounds are physically the same or physically different.
The usual phonetic treatment of post-voiceless "devoicing" in English words like slip, small, snake, shrimp etc is that there is gestural overlap between the spread glottis of the preceding obstruent and the sonorant phoneme, that is, even the treatment as [w̥] is linguist's over-phonologization of coarticulation – whereas in the case of "which", we have a clearly voiceless phoneme. Davenport & Hannahs are simply engaging in common over-phonologization of articulatory detail.