Static classes defined inside another class are called static nested classes, while non-static ones are called inner classes. Static nested classes can have static members, while inner classes cannot -- so the FAQ is correct as it stands. This is an important distinction to make. Sun's own documentation was sloppy on this at first but in recent years has been very clear.
Those terms are not wrong, but they're also not the "official" terms, the ones that appear without explanation in Sun documentation, or on the exam. It definitely pays to know the official definitions.
I disagree. The actual SCJP generally avoids such terminology in favor of examples. As for other documentation and definitions, "member class" is a well-defined term in the JLS (along with "member interface" and "member type" of course). And whether it's static or not is also well-defined. Err, maybe I should say it's strictly and unambiguously defined. Sun, in the past, made some poor choices as to how to communicate this stuff.
The term "static member class" means exactly the same thing as "static nested class" - if only because there's no such thing as a static nested class that is not also a member class. However the terms "inner class" and "non-static member class" are not the same thing - the latter is merely a subset of the former. The former also includes local and anonymous classes, while the latter does not. M&R use the term "non-static member classes" deliberately, for cases where they are not talking about local classes. The JLS refers to inner classes deliberately, for cases where they are including local classes. And when the JLS doesn't want to include local classes, they talk about "member classes", much like M&R.
Bottom line: each of these terms is well-defined by the most authoritative Sun sources (now). Some of these terms mean the same thing, and some mean different things. All of these terms are things that, in my opinion, a well-informed programmer should understand.