I have always had a hard time explaining to my students the term one-to-one. After making sure my students understand "in", "sur" and "bi", the Bourbaki terms, injective, surjective and bijective make sense to them. But to me, one-to-one sounds like it should mean bijective, not injective. I sometimes tell my students that injective should be called two-to-two, meaning that two different elements are mapped to two different elements.
ONE-TO-ONE CORRESPONDENCE is found in H. G. Zeuthen, "Sur les points fondamentaux de deux surfaces dont les points se correspondent un à un," C. R. LXX. 742. (1870).
One-to-one correspondence is found in English in 1873 in Proc. Lond. Math. Soc. IV. 252: "The equations .. being supposed to establish a 'one-to-one' correspondence between the two integral spaces." In his Principles of Mathematics Bertrand Russell (1903, p. 113) states, “Two classes have the same number...when their terms can be correlated one to one, so that any one term of either corresponds to one and one only term of the other.” (OED).
It sounds to me like Russel is talking about a bijection. Who and when started using "one-to-one" to mean injection?