The words "injective" and "surjective" are not that old. See here:
INJECTION, SURJECTION and BIJECTION. The OED records a use of injection by S. MacLane in the Bulletin of the American Mathematical Society (1950) and injective in Eilenberg and Steenrod in Foundations of Algebraic Topology (1952). However the family of terms is introduced on p. 80 of Nicholas Bourbaki’s Théorie des ensembles, Éléments de mathématique Première Partie, Livre I, Chapitres I, II (1954). Reviewing the book in the Journal of Symbolic Logic, R. O. Gandy (1959, p. 72) wrote:
Another useful function of Bourbaki’s treatise has been to standardise notation and terminology… Standard terms are badly needed for “one-to-one,” “onto” and “one-to-one onto”; will Bourbaki’s “injection,” “surjection” and “bijection” prove acceptable?
The terms did prove acceptable, even to mathematicians writing in English, and quickly became standard. For instance, all three terms are used in Jun-Ichi Igusa “Fibre Systems of Jacobian Varieties,” American Journal of Mathematics, 78, (1956), 171-199. (JSTOR search) The adjectival forms appear in C. Chevalley, Fundamental Concepts of Algebra (1956): “A homomorphism which is injective is called a monomorphism; a homomorphism which is surjective is called an epimorphism.” (OED)
You are certainly right that one cannot immediately derive from the word injective that distinct elements of the domain are mapped to distinct elements of the range. But in my opinion the colloquial use of "inject" suggests that distinct things remain distinct if they injected somewhere (for example molecules during an injection of a substance).
Anyway, we should accept that mathematical terminology is a specialist language allowing a lot of creativity. One should not try to overinterpret words used here.