I see some prior questions talk about the difference between one-to-one mapping and one-to-one correspondence.
One-to-one mapping vs one-to-one correspondence
One-to-one mapping – Injection
One-to-one correspondence – Bijection
However from the book "First Course in Abstract Algebra" by John Fraleigh says that 2 sets X & Y have the same cardinality if there exists a one-to-one function mapping X onto Y, that is, if there exists a one-to-one correspondence between X & Y. The "that is" in this sentence seems to be saying that both are the same thing.
Best Answer
The critical word in what you wrote is "onto". If $f:X \to Y$ maps $X$ onto $Y$ it usually means that this map is surjective, hence if it maps onto $Y$ and is a one to one mapping, its bijective, as well as when it is a one to one correspondence. Therefore both of the definitions you gave are basically the same, the terms are just confusing. I hope this helps you.