No, they are not equivalent.

It is a nice theorem that if there exists an infinite Dedekind-finite set (which is a set which satisfies the negation of (2)), then there is one which satisfies the first condition.

If $D$ is a Dedekind-finite set, then $S(D)$ which is the set of all injective finite sequences from $D$ is also Dedekind-finite (because the sets are injective, every collection of them is uniformly enumerated, so if there was a countable infinite set of these sequences, their union would be a union of uniformly enumerated sets, which would be a countably infinite subset of $D$).

Now simply consider the projection from $S(D)\setminus\{\varnothing\}$ onto $S(D)$ where you remove the last coordinate of the sequence.

You might be interested in the following paper:

Truss, J. **Classes of Dedekind finite cardinals.** *Fund. Math.* **84** (1974), no. 3, 187–208. PDF

In which the author takes seven definition of finiteness (proposed by Azriel Levy) and investigates the relations between them. Your first condition is one of the properties considered there.