This is more of a naming convention question than anything else. I am reading up on spectral sequences mainly from the book Lecture Notes in Algebraic Topology by Davis and Kirk. Now in this book the existence of what is labelled as the Leray-Serre-Atiyah-Hirzebruch Spectral Sequence is stated as:
Theorem: Let $F \to E \to B$ be a fibration with $B$ a path-connected CW-Complex. Let $G_{\star}$ be an additive homology theory. Then there exists a spectral sequence $$H_p(B, G_q(F)) \cong E^2_{p, q} \implies G_{p+q}(E).$$
The authors, Davis and Kirk refer the reader to the book Elements of Homotopy Theory by George Whitehead for the construction of the Leray-Serre-Atiyah-Hirzebruch Spectral Sequence. However in Whitehead's book the Leray-Serre-Atiyah-Hirzebruch Spectral Sequence seems to just be called the Spectral Sequence of a Fibration.
From skimming through Whitehead's book and Allen Hatcher's book Spectral Sequences in Algebraic Topology it seems that the Serre Spectral Sequence is just a special case of the Leray-Serre-Atiyah-Hirzebruch Spectral Sequence when we take $G_{\star}$ to be ordinary homology with integer coefficients.
However it Hatcher's book (on page 18 under "Generalized Homology Theories), he mentions that one can generalize the Serre Spectral Sequence to arbitrary cohomology theories. The spectral sequence attained which he doesn't give a name to, I assume, would be the Leray-Serre-Atiyah-Hirzebruch spectral sequence above. He then goes on to say that the Atiyah-Hirzebruch Spectral Sequence is a special case of this unnamed spectral sequence in the case when the fibre $F$ is a point.
My assumption seems to be further validated by the Wikipedia article on the Atiyah-Hirzebruch Spectral Sequence which states that Atiyah and Hirzebruch found an (unnamed) generalization of the Serre Spectral Sequence (which I again guess would be the Leray-Serre-Atiyah-Hirzebruch Spectral Sequence above) of which the Serre Spectral Sequence is a special case when one takes singular homology (or since singular homology is isomorphic to ordinary homology with integer coefficients for CW complexes). Further still it states that the Atiyah-Hirzebruch Spectral Sequence is a special case of this unnamed spectral sequence when the fibre $F$ is a point.
It seems then that both the unnamed spectral sequences mentioned above are what is referred to in Davis and Kirk's book as the Leray-Serre-Atiyah-Hirzebruch Spectral Sequence, is this correct? Furthermore is the Leray-Serre-Atiyah-Hirzebruch Spectral Sequence equivalent to the Spectral Sequence of a Fibration?
If so then what is the naming convention for this spectral sequence, is it usually called the Spectral Sequence of a Fibration as in Whitehead's book or the Leray-Serre-Atiyah-Hirzebruch Spectral Sequence as in Davis and Kirk's book. In other words which name is most often used to reference this spectral sequence in books, articles, papers etc.?