To add to @KeithInc., while there are many uses for the term, the most common use of “cistern” has always been “rainwater collection/storage tank”. Sure, it was very common for “water closets¹” to draw their water from a cistern in that sense, especially prior to modern plumbing, but so did the bathtubs and sinks. I suspect they’re thinking cistern means “septic tank”, which says more about them than anything else. [Correction, see below]
¹From what I’ve been able to find, “toilette” originally mean “changing-room table” in the mid-1500s, then it later meant “hygienic cleaning cloth” in the early-1700s, and then became a synonym for “washroom” (and the spelling changed to “toilet” in English) in the mid-1800s, while the “water closet” was a separate section for the “flushing privy / flushing latrine” at first, before referring to the privy/latrine itself… around 1920 the word “toilet” (washroom) started commonly being used to refer to the “water closet” (toilet) and the room itself started being called “restroom” and/or “bathroom”. That’s a long, strange linguistic journey.
[Edit: After posting this, it occurred to me that what I know as a “toilet tank” might technically be called a cistern. After checking, apparently it is. Scratch the part about septic tanks, but it’s still dumb… who even thinks that? I’m not sure whether most people even know that usage.]