Let’s have a look at the specific f84r bathing section (again):
I was reading about bathing in general in medieval times and i stumbled upon the word “zistern” (German) or cistern.
It seems that during 7000 years these are being used, mostly underground, for storage of (drink)water. In bathing areas they consumed a lot of water and they needed a bassin for the storage of clean water. For that purpose alone often cistern’s were made.
If you google on: cistern medieval underground
you will see there are a lot of sites that are quite interesting. There even is a list of those medieval underground cisterns on wikipedia: https://en.wikipedia.org/wiki/Cistern
and go to “examples”.
You can also find the paper: History of Water Cisterns: Legacies and Lessons Larry Mays 1,*, George P. Antoniou 2 and Andreas N. Angelakis with quite some interesting details and links to ancient cisterns. For example: http://www.minoancrete.com/tylisos.htm
Although these cisterns were not directly used for bathing, we can easily imagine that they could be used (secretly) for a private bathing.
Bathhouse cosmetics: https://nl.pinterest.com/gingerbatsy/medieval-bathing-cosmetics/