Is there an accepted definition of coprimality in commutative ring theory?

Question

I can think of at least three possible definitions of coprimality in commutative ring theory: call $a,b \in R$ are coprime iff

1. if $c \mid a$ and $c \mid b$, then $c \mid 1$.
2. if $a \mid c$ and $b \mid c$, then $ab \mid c$.
3. $1$ can be written as a linear combination of $a$ and $b$, and hence every $r \in R$ can be written as a linear combination of $a$ and $b$.

These all agree for $\mathbb{Z}$.

I remark that (1) makes sense in an arbitrary poset, (2) makes sense in an arbitrary commutative monoid, and (3) makes sense in an arbitrary commutative ring.

Question. Is there an accepted definition of coprimality in commutative ring theory? If not, is there at least an accepted definition in principal ideal commutative rings?

Addendum 0. Here's another possible definition: call $a$ and $b$ coprime iff for all $a_0,a_1 \in R$ that divide $a$, and all $b_0,b_1 \in R$ that divide $b$, we have: $$a_0b_0 \sim a_1b_1 \rightarrow a_0 \sim a_1 \wedge b_0 \sim b_1$$

Addendum 1. Here's another one: call $a$ and $b$ coprime iff for all $a',b' \in R$, and all $r \in R$, we have that if $r(a',b') = (a,b),$ then $r$ is a unit.

Addendum 2. Darnit, here's another one: call $a$ and $b$ coprime iff $a$ is a unit in $R/bR$.