I think that the reason for this old terminology is that the people inventing schemes for the first time they were guiding themselves with what Serre had done already in FAC. There Serre defines first an algebraic prevariety as a ringed space which is locally isomorphic to an affine variety (“affine variety” here is a Zariski closed subset of $\mathbb{A}^n_k=k^n$, equipped with the sheaf of regular functions, i.e., functions which are locally given as a quotient of polynomials) along with the additional size restriction of quasi-compactness. After, he defines an (abstract) algebraic variety to be a prevariety that is additionally separated (for the definition of this notion of separatedness, which is not the exactly same as for schemes, see the paper itself from Serre, Chapter II, nº 34; or for a more modern treatment, see Milne's book, Chapter 5, sections c and h).
[What follows is a guess of myself, and not a completely informed historical comment. I would appreciate if someone corrected me if there is something inaccurate in the following ideas.]
The people that were inventing schemes generalized prevarieties to preschemes, as ringed spaces that are locally isomorphic to an affine scheme, and then generalized algebraic varieties to schemes, as preschemes that are separated (in the modern sense). My supposition is that people realized that once one achieves such a broad generalization there is no need to fixate oneself with an additional axiom of separation, and the true interesting object of general study is what previously was called a “prescheme” and today is called a scheme.
I don't know how much time it took mathematicians to move on to the new terminology. As the OP points out, Mumford's Red Book, that was published in 1967, still used then the old terminology. Around 1971 it seems that the new terminology was applied already by Grothendieck:
(Taken from here.)