The article you linked literally says that there are two definitions: one for real numbers and another for complex numbers. Thanks for the info.

That’s not what it says. It says that there is a single definition that can be generalized to both real and complex numbers.

A special cases of the general definition where im(z)==0 yields an expression where some parts are multiplied by zero, and can then be omitted entirely.

This means that there is one definition. You can mentally ignore some parts of this when dealing with reals.

There is one definition: the distance to 0. There are several (more than two) different ways to calculate it in different situations.