What is the better definition of absolute value should be i give to my students?

Question

The absolute value has two distincts definitions as shown below :

Definition 1: $|x|=\begin{cases} x, & \mbox{if }x \geq0 \\ -x, & \mbox{if }x< 0 \end{cases}$

Definition 2: $|x|=\begin{cases} x, & \mbox{if }x \geq0 \\ -x, & \mbox{if }x\leq 0 \end{cases}$

As shown here the definition 2 is the generalisation for ordered ring and it used widely in real analysis , but the definition 1 is rarely used at teachers and students , then my question here is :

Question: What is the better definition of absolut value should be i give to my students ?

Note: I prefer to use the definition 1 to avoid repetution of $0$ in two sides

Answer 1

The two definitions say exactly the same thing. This is also the same: $$|x|=\begin{cases} x, & \mbox{if }x >0 \\ -x, & \mbox{if }x\leq 0 \end{cases}$$

Your students should, most importantly, understand that $|x-y|$ is the distance between $x$ and $y$ on the real line.