Is a cistern under a hamam the same as a bathing pool?
No. A cistern stores water; it is not automatically a pool where visitors bathed. Its purpose and date need to be established for that particular site.

Related questions
Are there Byzantine remains beneath Istanbul hamams?
At Zeyrek Çinili Hamam, Byzantine cisterns survive beneath the Ottoman bath. That is this building's story; it does not describe every hamam in Istanbul.
Why does restoration sometimes uncover older remains?
Work can reveal earlier layers concealed beneath or behind later structures. Archaeological findings need documented interpretation, rather than assumptions based on appearance.
Does an old-looking hamam have to be historic?
No. Marble, domes and Ottoman-style decoration can also be modern. Look for documented construction and restoration dates if the age of the building matters to you.