As you have already stated the scriptures recommend taking bath either in rivers or in water that is drawn out from wells.
The river water is considered pure as it is always flowing ( i.e because of the current). And, the water in the well is considered pure because it is always in touch with Bhumi (or Earth which is a Vedic deity).
From Dharma Bindu (which has quotes from various scriptures) we get lot of valuable info on Snana Vidhi:
Vyasa opines: Praatah kaaletu sampraapte kritvaa chaavashyakam
Snaayaannadeeshu shuddhaasu shoucham kritvaa yadhaa vidhi/
A virtuous person would perform his morning routine and then take
bath in rivers or running waters.
But in today's time it is not possible for most of us to follow such instructions. So, we have adjust with whatever we can manage to do.
The book further quotes Manu and other authorities and also gives a shortcut for compensating for not bathing in rivers. It says when we are using stored water for bath first discard seven palmfull of water from it and then bath. Then it is acceptable even by these ancient standards.
Nadeeshu Devakhaateshu Tataakeshu Sarassucha snaanam samaacharennityam
Manu says that Snaanas be taken among the water bodies like Jeeva
Nadis or ever flowing rivers, divine water bodies like Pushkar and
Manasarovara, and wells and deep water pits.
Puraanaanaam Narendraanaam rishinaamcha mahatmanaam,
kupatataakeshu devataanaam samaacharet/
Tatopi Saarasam punyam Gaangam punyantu
Markandeya describes that in the past there were some water bodies
dug up in which Kings, Rishis ans Mahatmas bathed and performing
snaana at such places are worthy indeed. Water dug up from the depths
of earth is sacred and so are the water falls and of flows; reputed
Sarovaras and Rivers like Ganga are highly sacred.
Yoga Yagnyavalkya prescribes that whenever one has to take bath elsewhere, then snaana be done by discarding palmful of water seven
times first and then proceed. Also, wherever there is a running
River of Sacredness nearby, one should prefer bathing in it rather
than in any stored water!
Or else one can also invoke the presence of Ganga, the most holy river, by chanting the Sloka given in this answer.
nandinI naLinI sItA mAlatI cha mahApagA
bhAgIrathI bhogavatI jAhnavI
dvAdashaitAni nAmAni yatra yatra jalAshaye
snAnodyataH smarennityaM tatra tatra vasAmyahaM ||
Manu Smriti 5.134. In order to cleanse (the organs) by which urine and
faeces are ejected, earth and water must be used, as they may be
required, likewise in removing the (remaining ones among) twelve
impurities of the body.
Thus,in ancient times purification was attained by the use of earth/soil and water. In today's time the role that earth used to play is played by soaps etc. Also oil bath is mentioned and allowed in scriptures so there is no reason why using soap/shampoo will be bad.
Moreover, the water that we get from overhead tanks are collected from some river or the other. And, water is always pure as even the most impure substances can not sully it.
A woman is not polluted by a lover, a Brahmin by Vedic rites, water by
urine and faeces and fire by the act of burning. (Vashishta Smriti
Therefore, there is nothing to worry about.
But if we want then we can employ either of the two techniques described above - the discarding seven palmfull of water technique or the invocation of the holy rivers technique to make our bathing procedure meet the scriptural standards.
The book Dharma Bindu is available on Kamakoti.org and can be downloaded from here: https://www.kamakoti.org/kamakoti/articles/Dharma%20Bindu.pdf