What product or method to waterproof an underground concrete-bottomed water cistern?
From the potability standpoint I think you will want to definitely plan to have high performance hand operated water filter unit available to filter any water you would try to drink from this tank. Check at a place like REI to find a filter like the type used by canoeists when they make trips on rivers and lakes and filter lake water for drinking.

It seems to me that it is going to be very difficult to get this tank dry enough to seal it. One way that may work is to dig down five or six holes that are  about a foot in diameter to a depth of maybe 13 to 14 feet deep (i.e. a good amount below the existing concrete tank floor). Then insert a sump type pump into the bottom of each hole that pumps out accumulated water as fast as it can accumulate. You would want to spill the pumped water out in the street or rain sewer so as to carry the water away from your cistern area. It is possible that this could lower the water table near the tank bottom enough to keep it dry for a period of time. 

Of course you would want to do this at a time of the year when there has been little or no rain for a goodly length of time.