Hi Speedemon311,
If there is the Rook symbol, followed by "ab1", then you're correct in assuming it means the Rook on the a-file moves to the b-file along the first rank. When you say this move doesn't match the board diagram, are you referring to page 8, second paragraph? Each time the two players' names are mentioned, that is where the new board diagram should be taken into consideration. So when Soltis says "Here 20. e5, 20. Rab1..." he is actually still discussing the diagram on page 7, not the first diagram on page 8. Hope this helps.
I have had the book "How to choose a chess move by andrew soltis" for a few weeks now and can not read the notation. Its sapost to be "figurine algebraic chess" notation but it isnt. Even on the first pages, with no explanation, it has stuff like the little white brook simbol then ab1 behind it. So I thought maybe its saying the brook goes from a to b on the first row of the board which is imposible with the shown picture. The book is littered with this stuff. or Brook ad1 which is also imposible baised on the picture of the pieces. Driving me crazy!