In the definition of an ideal there is stability by multiplication by any element of the ring $$ri\in I\ \forall r\in R, i\in I$$
In particular we have the internal multiplication stability because $r\in I\subset R$.
In the sub-ring definition we only have stability by internal multiplication. $$s_1s_2\in S\ \forall s_1,s_2\in S $$
An ideal is a sub-ring, that I understand... How come the ideals of a ring different from $R$ are precisely the sub-rings that don't contain $1_R$ ?
Edit:
It is not true that a non-unital sub-ring is necessarily an ideal. I misinterpreted a remark, there is a counter example in the comments. Sorry about that.