Is a set of orbit representatives same as a set of coset representatives?

Question

Let G be a multiplicative abelian group and H $\subset$ G. Assume H acts on G by multiplication.

1. Is a set of orbit representatives of G under the action of H same as a set of coset representatives of H in G? If so, is G/H the correct notation for both the sets?

2. If G is a semi-group instead, is it legit to talk about a set of coset representatives of H in G? (As far as I know of, coset representatives arise from the equivalence relation defined on a subgroup of a 'group'.)

Answer 1

1. Is the set of orbit representatives of $G$ under the action of H same as the set of coset representatives of $H$ in $G$?

Yes, but rather than write the set of orbit representatives, it is better to write a set of orbit representatives, since the representatives are not unique.

If so, is $G/H$ the correct notation for both the sets?

$G/H$ is the notation for the set of cosets. It is in bijection with any set of coset representatives, but not equal to a set of representatives.

If G is a semi-group instead, is it legit to talk about the set of coset representatives of H in G? (As far as I know of, coset representatives arise from the equivalence relation defined on a subgroup of a 'group'.)

It is not legit to talk about the cosets of a subsemigroup $H\leq G$. But it is legit to talk about the cosets of a semigroup congruence.