This is a feature of certain Hakka varieties, but there is some controversy. Jerry Norman's 1986 work "What is a 客家 Kèjiā Dialect?" puts it thus:
To determine whether a dialect is Kèjiā or not, one should examine shǎng tone words having sonorant (nasal and lateral) initials. If the dialect is truly Kèjiā, such words will fall into two groups, one in the yīnpíng category, and the other in the yīnshǎng category; the incidence of one tone or the other will be determined lexically.
In essence, those that have been merged with yīnpíng or "dark level" will be those more likely to be used in colloquial speech. Note that this is related to, but slightly distinct from, the phenomenon of colloquial vs literary readings (文白異讀): from the list you have above, 武 and 柳 show this latter phenomenon, and I would expect compounds with the "dark level" reading to be colloquial in nature. However, looking through moedict.tw, the evidence is ... inconclusive.
This lack of consistency is only exacerbated when you expand the scope past the Meixian 梅縣 / Taiwan Sixian 台灣四縣 varieties of Hakka. Sagart's 1998 work On Distinguishing Hakka and Non-Hakka Dialects has a table which outlines the development of twelve "diagnostic words" that were cited by Norman: but not all the five Hakka dialects cited split these the same way (although there is substantial debate on how many of these are 'actually' Hakka e.g. Is the dialect of Huizhou 惠州 in Guangdong Province a form of Hakka or Yue?).
For Dayu, Huizhou and Heyuan, very few 次濁上聲 (voiced sonorant rising tone) 'colloquial' ("set 1" according to Sagart) lexemes merge with 陰平 ("dark level"), and instead merge into 陰去 ("dark departing"). Ninghua and Ganxian don't show much literary-colloquial split in these lexemes and merge most of them into the one 上聲 category. Huizhou is even reported as having 'literary' ("set 2") lexemes allocated to 陰平 ("dark level"), rather than its 'colloquial' ones.
The Yue variety of Taishan (Toisanese) also has some allocation of 次濁上聲 (voiced sonorant rising tone) lexemes to its 陰平 (dark level) tone category as well; this is mentioned in a previous answer on Taishanese - Gan connections. Sagart (1998) states that the lexical incidence of those that merge into 陰平 ("dark level") [e.g. 老] vs those that stay in 陽上 ("light rising") [e.g. ] vs those that merge into 陰上 ("light rising") [e.g. 女]:
presents important similarities with the lexical incidence component of [the criterion].
Of course, one can see that the precise lexical incidence does differ from Meizhou / Meixian Hakka.
The emergence of this particular 陽上 "light rising" to 陰平 "dark level" tone reallocation for Hakka but also for Nanxiong/Xiongzhou dialect (another Hakka-Yue border topolect) is detailed in Sagart (2001) Nanxiong and Hakka, positing that the tone reallocation started with voiced fricatives first, then Nanxiong split off and stopped that change there, whilst the Hakka of southern Jiangxi continued to allocate other voiced obstruents and also voiced sonorants to 陰平 "dark level". According to Norman (1986:340), and reiterated in Coblin's 2019 Reconstruction of Common Neo-Hakka (where Tone 1 = 陰平 "dark level"):
[O'Connor and Norman] reasoned that the sonorant initials of such words must have differed from the ordinary voiced ones present in the large body of "regular" syllables now found under Hakka Tone 1. And their explanation for this was that these aberrant sonorants were in fact voiceless at an earlier historical stage.
Hence the merger of 陽上 "light rising" to 陰平 "dark level" is seen by them as the 'regular' development, with the reflexes in the merged 去聲 "departing" being a literary import, related to borrowings from "a northern dialect or koine" in "late reading traditions", and reflexes in the merged 上聲 "rising" also being similar. However, these 濁上歸去 must have happened, according to Coblin, when this northern literary superstratum...
still retained upper and lower register qùshēng tones and assigned Common Dialectal Chinese obstruent initial yángshǎng words of this type to the lower tone (i.e., Tone 6, yángqù).
