Is there a term for a magma with identity (only)?
The term "unital magma" does get used, including by Bourbaki (Algebra I, p. 12):
A magma with an identity element is called a unital magma. Here, an identity element is defined to be two-sided.
Divisibility does not imply invertibility, so they are distinct. One can have divisibility without an identity element; one needs both for invertibility. Note: a left inverse and a right inverse might not be the same, but invertibility does not demand this.