[0PingWithJesus]

The detector does not "look" in any direction, it is in no way "pointed" at the sun. It records the direction of all events that occur within its volume. But once recorded they compare the direction of all events with the direction from the sun at the time of the event. The angle between the solar direction and the event direction is what makes up that image. If the neutrinos were not coming from the sun, the image would look like white-noise. Since there is a clear "peak" at the center you can make a good estimate about what fraction of events in your data set came from the sun. That amount is a direct measurement of nuclear processes going on with the sun over the course of the dataset...which is physically interesting. Here is the 1-D version of the neutrino "picture", https://i.imgur.com/7OmXXtn.png (cite: https://arxiv.org/pdf/1606.07538.pdf). You can tell quite clearly that there are many more events pointing away from the sun then are pointing back towards it. Exactly how much more is the interesting physics measurement done here.

All that being said, the specific shape of the "sun" in the image is influenced by many factors many of which are related to the detection mechanism and the detector itself...and don't tell you that much about the sun. Eventually (one hopes), detectors will improve to the point where the "shape" information of the image is reliable enough to extract interesting solar physics measurements from it.

P.S your fun thought on the detection of a fusion reactor is extremely on point. There exists a under-construction experiment in the UK called "Watchman" that hopes to detect a neutrino signature from a nuclear power plant being shut off and then being used to produce material for a nuclear weapon. The idea would be that you could observe activities of nuclear facilities in a "rouge nation". See here https://www.nytimes.com/2018/03/27/science/nuclear-bombs-ant... or here http://svoboda.ucdavis.edu/experiments/watchman/