Notion 2 was first considered for groups in Wedderburn, J. H. M. Homomorphism of groups. Ann. of Math. (2) 42, (1941). 486–487. But the main results show for groups the notion is not so exciting. In semigroup theory relations satisfying 2 are called relational morphisms. They form the key notion of morphism in finite semigroup theory. They were introduced by Eilenberg and Tilson in Eilenberg's book on finite semigroup theory.
For instance, one can consider which elements of a finite semigroup relate to 1 under all possible relational morphisms from to a finite group. The answer is the smallest subsemigroup containing all idempotents and closed under $x\mapsto axb$ whenever aba=a or bab=b. One proof uses that the product of finitely generated subgroups of a free group is closed in the profinite topology (which was conjectured by semigroup theorists interested in this problem and proved by Ribes and Zalesskii using profinite groups acting on profinite trees).
