i love your bog i think it's mostly evaporation since i actually lose about 0.5 inch yesterday and it was a very very hot day in the 95+ and high heat index. i dont know what that little wet spot is though. i think its a very tiny hole that water just seep through. i might try to use dye to see if it comes out.
The heat index is a measure of how uncomfortable (well, how hot) it feels because of the combined effects of the temperature and humidity. A high heat index relative to the temperature means that the air is very humid. A high humidity level will actually decrease the evaporation rate because the air is already so saturated, it absorbs water more slowly. It is, of course, still very hot though, but
Just to make sure I'm understanding this correctly. You lose 2" a day when the pumps and stream are running, correct? Now with the pumps and stream off, you lost 0.5" in a day, right?
With it being such a new pond (without a heavy plant load to suck up water), and now without running water, 1/2" sounds excessive to me. Did you set out a pan of water to measure evaporation as well, to use as a control?
I found a study on evaporation rates from around the country. http://www.nws.noaa.gov/oh/hdsc/PMP_related_studies/TR34.pdf
I'm not sure where you're located in NC, but the highest monthly evaporation rates were 7.4" in June. This is an average of less than 1/4" per day. Now, of course, running water will increase this rate, but still and running water rates of 0.5" and 2" per day, respectively, seem excessive to me. I wouldn't be satisfied with calling that evaporation if it were my pond.
Besides a possible leak in the bog, I'd assume there is also a leak somewhere in the stream or pump system, since an extra 1.5" per day is lost when the pumps are running. Keep in mind, with your sized pond (I think you said 12x15', right?), 2" drop per day is 225 gallons a day. That's a lot of water.
Try drying the backside of the liner with a towel. Then keep a towel behind the wet spot for a period of time and then see if you can more accurately pinpoint where the water is coming from, based upon what part of the towel is wet.