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Concerning the local class field theory normalization, the real reason Deligne’s normalization is called what it is and is the good one is that if X is a scheme over a finite field k of size q then on

(*) X \otimes_k \overline{k}

the composition of 1 \otimes Frob_k and Frob_{X/k,q} \otimes 1 is (obviously) the absolute q-Frobenius of the F_p-scheme (*) and (not obviously!) it is a general fact for any separated finite type F_p-scheme S that pullback along the absolute Frobenius of S (and hence any iterate) induces the *identity* map on every H*(S,M) and H*_c(S,M) with constant coefficients M. This is a special case of a general fact that is explained in more detail on pages 89-91 of my etale cohomology notes

http://math.stanford.edu/~conrad/Weil2seminar/Notes/etnotes.pdf

The point is that therefore the effect of pullback along 1 \otimes Frob_k^{-1} on such constant-coefficient cohomologies of (*)  is the *same* as the effect of pullback along Frob_{X/k,q} \otimes 1, and it is the *latter* that has genuine geometric meaning via coming from an actual k-scheme map X —> X which is what actually comes up in Lefschetz trace formula (and has eigenvalues with good integrality properties).  Ergo, “geometric Frobenius”.

In other words, the Galois element Frob_k^{-1} is the number theorist’s way of getting the effect of the algebraic geometer’s Frob_{X/k,q} as far as endomorphisms of etale cohomology is concerned.  If one thinks geometrically then this is all about the q-power map on X or its scalar extension to \overline{k}.  But if one wants to get the same effect through an element of Gal(kbar/k) (which an algebraic geometer may not want to do but a number theorist would) then it happens to coincide with the effect of Frob_k^{-1}. 
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