Thank you! Yes, M^* is Stewart's notation. Indeed this notation is "lossy" in the sense that it loses information about what the extension field is. The extension field is usually introduced once for every example or theorem and then M^* is used as a shorthand notation. For example, in my post, the extension field is introduced like this:

"The notation M^* denotes the group of all M-automorphisms of L with composition as the group operation."

By the way, Stewart uses the notation Γ(L/M) too at several places but in this specific lemma, the notation M^* is used and therefore my post too uses the same notation.