In Lenstra’s notes on Galois theory for schmes, he defines a Galois category as a category satisfying a variety of properties (page 33) together with a functor $F$.
One of these properties is that
“The functor $F$ transforms terminal objects in terminal objects and ... transforms epimorphisms in epimorphisms.”
The usage of the word “in” is throwing me off. Does this just mean that if $A$ is a terminal object in the category $\mathcal{C}$ that the object $F(A)$ is also terminal?
I am pretty certain your guess is right. The terminology is indeed a bit nonstandard (from a category-theoretic point of view). Usually we would say the functor "preserves terminal objects".
The reason I think you are right is because the precise wording in the document is:
Then this statement is equivalent to saying that $F$ preserves all finite limits (the more common category-theoretic term for "fibred products" is "pullbacks"), something that is common to ask of a functor. Such a functor is also called left exact, on the nLab page you can find more on these left exact functors. Including the claim about the equivalence I just mentioned (Proposition 3.2 on that page, at the time of writing).