The Sanskrit word ‘dhauta’, ‘cleansed,’ by itself, does not mean ‘cloth’ and much less ‘dhoti’. In fact, in the entire Vedabase, as well as in the Sanskrit dictionary, the word dhauta is only used as an adjective, not as a noun. And the Sanskrit noun ‘dhauti’ means ’the act of washing’.
In our sacred texts, the word dhauta is always used as an adjective meaning ‘cleansed’, and does not especially refer to cloth.
Srimad Bhagavatam references:
2.8.6 dhauta-atma = purified soul
6.4.4 dhauta-anghri-panir = cleansed feet and hands
6.18.52 dhauta-vasa = washed cloth
10.12.38 dhauta-pataka = cleansed of sin
10.53.44 dhauta-pada-kara = washed feet and hands
11.17.23 dhauta-dad-vasah = cleaned teeth and cloth
11.27.10 dhauta-dantah = cleaned teeth
Note: we see at 6.18.52 and 11.17.23 that ‘dhauta’ does not by itself mean cloth. The word ‘cloth’ must be added. We find the same in Santana Goswami’s Brhad-bhagavatamrta 1.6.12 dhauta-anga-ragah = [Krishna’s tears] washed away his body cosmetics.
Sri Caitanya-caritamrta references:
Adi 4.170 dhauta-vastre = cleaned cloth
Past Acharya references:
Here are more examples from Bhaktivinoda Thakura’s writings, showing that even as late as the 19th century, dhauta by itself does not mean cloth, much less ‘dhoti’.
Gitavali sarva-dosa dhauta kori = cleansing all impurities
Hari-bhakti-kalpa-latika 8.10 jala-dauta-padah = feet washed with water
Hari-bhakti-kalpa-latika 10.19 dhauta-samasta-mano-malah = cleansed of all mental impurity.
Jaiva-dharma dhauta-vastra-dharah = [Krishna] wears clean cloth