How did "one-to-one" come to mean "injective"?

In the old usage, as well as contemporary usage in set theory, one may consider a function without specifying a particular codomain or target set. (The insistence that a function come along with a particular codomain is a comparatively recent innovation, probably arising in Bourbaki.)

That is, if one understands a function merely to be a set of ordered pairs satisfying the function property (that each input is associated to one output), or as a rule associating to every object in a domain an output value, then it is true to say that a function is one-to-one if and only if it is a bijection from its domain to its range. Thus, injective functions really are one-to-one in the sense that you want.

Of course, this one-to-one terminology was long established by the time Bourbaki wanted to insist that functions come along with a specified co-domain, giving the definition of function as a triple consisting of domain, codomain and set of ordered pairs. The fact that in this context the concept of one-to-one doesn't tell the whole story may be part of the reason that they introducted the injective, surjective, bijective terminology.

But meanwhile, a function is one-to-one if and only if it provides a one-to-one correspondence between its domain and its range. This is perfectly logical, and seems to be the explanation that you are seeking. I would think that the one-to-one terminology begins to seem illogical only when one also insists on attaching to the function a target set or codomain that is not the same as its range, which is, after all, a somewhat illogical thing to do.

The terms injectif, surjectif and bijectif are lexical creations of Bourbaki. Their first appearance was in Chapters I and II of his Théorie des Ensembles, published in 1954. (Mac Lane and others knew about them and had used them in print slightly before)

Surjective functions were called fonctions sur in contradistinction to general functions, just called fonctions dans. Bourbaki was very attentive to the quality and beauty of the French he used and found it shocking to use the preposition "sur", instead of a genuine adjective, to qualify the name "function". Hence the neologisms.

Here is a link, in English, to this theme (look at the entry "Injection, surjection and bijection").

As to the terminology one-to-one, your actual question !, the reference I give attributes its first use to Zeuthen in 1870 (in French). The first appearance in English dates from 1873. There is also a reference to its use by Bertrand Russell in 1903.

The above doesn't really answer your question on the reasons why mathematicians used the terminology "one-to-one" but has the advantage of giving hard facts. As to these reasons, the suggestions and guesses in the comments look very reasonable and informed and I second them. And, by the way, your statement that one-to-one predates injective is now proved to be absolutely correct.