Same set of carboxylate reagents yielding two different products with an alkylide
Reaction 2 is wrong. Reaction 1 is correct.

Later in the book (q87), another question is asked with the same set of reagents (carboxylic acid and methyllithium). Reaction 1 has been used in that as well.

Reaction 2 is wrong because, as ralk912 noted above, charge conservation has to be followed. If methyl lithium reacts with a negatively charged species, the product has to be negatively charged as well. If the ester is neutral, then lithium must be negatively charged, and would associate with the $\ce{Li+}$ formed in step E to form $\ce{Li2}$, which is only observed in gas phase. 

Not to forget that there simply isn't any mechanism wherein a carboxylate would react with an alkylide to form an ester. It does not simply occur. Reaction 2 is wrong.



Waylander has suggested a good reason for a misprint i.e. the reagent $\ce{CH3Li}$ might have been intended to be $\ce{CH3I}$ instead. Then, it would make sense for the carboxylate anion to attack an alkyl halide, and displace an iodide ion (which could easily attach with the $\ce{Li+}$ ion to form lithium iodide) forming the methyl ester.



user55119 has confirmed that the product of the step F is cyclohexylmethyl ketone, and that it is obtained by reaction 1 only. The book containing this is "Margret Jorgensen in Organic Reactions."