I don't know if there is a name for this. Note that this can never happen for a finite monoid which is not a group. The minimal ideal of a finite semigroup is globally idempotent and if the finite monoid is not a group, this minimal ideal does not contain the identity.
Afterthought. Your semigroup cannot have any non-identity idempotents if its globally idempotent subsemigroups contain the identity.