coprime vs relatively prime

Question

Since I am not a native english speaker and really couldn't find it: Which definition in a ring (of course not integers) belongs to which word:

Let $R$ be a ring (commutative with 1) and $a,b \in R$, then $a,b$ are called relatively prime resp. coprime if

1. For any $c \in R: c$ divides $a$ and $b \implies c \in R^\times$

or

2. The ideal generated for $a$ and $b$: $\,(a,b)$ is $R$.

Of course these two definitions coincide in PIDs but in other rings they obviously differ. The question is now, which is which?

An easy example where it differs would be the polynomial ring over 2 variables $R=K[X,Y]$ for some field $K$ and $a=X, b=Y$. They fullfill 1. but not 2.