Perhaps you are not understanding edge orientation correctly. First there are two particular groups that are often used as a basis for defining edge orientation: <U,D,L,R,F2,B2> and <U,D,L2,R2,F,B>. (The order the elements are listed within the angle brackets does not matter.) When basing edge orientation on these groups, using only the group generators (the moves listed within the angle brackets), the orientation of edges do not change (by definition, basically).
To find out if an edge cubie is correctly oriented for the <U,D,L2,R2,F,B> scheme, move the piece to its proper location using only the moves U, D, L2, R2, F, and B. If it ends up correctly oriented in its home position, then it was also oriented where it started from. Otherwise the piece was incorrectly oriented where it was at the start.
For instance consider the FR edge cubie at location UL (in either orientation). It can be moved to its home position using U F. If U F solves that edge, then it was correctly oriented at UL. If instead it's flipped, then it was also incorrectly oriented at UL.
If using the <U,D,L,R,F2,B2> scheme, then instead of using U F (since F is not a generator for that group), use L F2 (U U R R R also works).
In the "Post 1" section of Ryan's description, G1 is defined as <L2,R2,F,B,U,D>. But in the "Post 2" section, he changes G1 to be <L2,R2,F,B,U,D>. So this inconsistency might be one thing that's confusing you. Interestingly, Thistlethwaite's original paper didn't use either of these groups for G1. It used <U2,D2,L,R,F,B>.
If you are interested in seeing the original Thistlethwaite algorithm, use this link:
http://www.geocities.com/jaapsch/puzzles/thistle.htm
Hope this helps.
