More precisely, a system is consistent if it does not contain a contradiction.

What we mean by a contradiction is the truth of a statement of the form "x and not x", or, in Bynerma's terms, that there is a statement x such that both x and "not x" logically follow (from the axiomatic system). (Things that we (correctly) intuitively recognize as contradictions allow us to derive a statement of that form.) So, although Bynerma used slightly unusual language, she correctly expressed the content of the usual definition.

The Law of the Excluded Middle is not needed here. It says that statements of the form "x or not x" are true (provable even, according to Wikipedia).