Quote by Lucretius
Obviously something has gone wrong with my thinking. But I can't quite pin it down. Both clock in their respective stationary frames represent proper times. Applying the time dilation formula I get these results. I can't figure this one out…

The problem you are having comes from thinking the “real time” observed by the traveler is the same as “proper time”. Proper time, proper distances and proper speeds can only be defined with respect to (wrt) to some reference. When you say “Applying the time dilation formula I get these results” you were figuring proper time wrt the Earth frame. If you had used another frame (say something traveling the other way at 0.8c) wrt that frame you would get a different “proper time” result, because you would be basing your measurements from a different inertial reference frame.
Hence when you look at
http://en.wikipedia.org/wiki/Proper_velocity explained it surly is confusing and even shows apparent observations by a traveler of FTL. It is important to remember that is only an apparent observation – nothing that can be directly observed at any point during travel.
You describe a traveler ending his trip 8 light years from earth in 10 years earth time but in only 6 years on the traveler clock. Obviously the will seem like FTL to the traveler, but that is only a “Proper Velocity” not a speed the traveler could ever directly observe for real any time during the trip. The wiki info has some additional links that might help.