Had you said that i wouldn’t have quibbled. 8, 10... pretty close. You said the probe operates at 8 watts.
Now to your other question. If you see pictures of New Horizons you see that it has a parabolic dish. That is so that most of the transmitted power is aimed in a narrow cone towards the Earth.
To answer your question we would need to know how wide the beam is. But it’s safe to say that they make the cone big enough to account for the accuracy with which they can aim the antenna at Earth.
Look, I agree. It is hard to understand how a 10 watt signal can travel billions of miles and still be picked up on Earth when an AM station sends 50000 watts and barely makes it 100 miles. It turns out we are that good. Using large antennae, serious physics, a quiet part of the spectrum, and a whole bunch of math, we can do it quite easily.
One other intersting tidbit. The probe acquired data for about 16 hours during the encounter. It will take 12-16 months to transmit it all back to Earth. That tells us something about the bit rate... :-)