I have seen two definitions of scale parameter of a distribution: Let $X$ be random variable and ${{P_{theta, X }: theta in Theta subset mathbb{R}}}$ be a parametric family of distribution of $X$ indexed by $theta$

$$1)~~ theta text{ is a scale parameter iff} rightarrow text{ the distribution of } cX text{ is in } {{P_{theta, X }: theta in Theta subset mathbb{R}}} text{ for every positive constant c}$$

For example Let $X$ ~ $LogNormal(theta, 1)~, theta in (-infty, infty)$ then $cX$ ~ $LogNormal(theta^*,1)$ where $theta^* = theta + ln(c) in (-infty, infty)$. With this definition $theta$ is a scale parameter.

For the second definition it is required that $theta$ is positive:

$$2) ~ theta > 0 text{ is a scale parameter } iff rightarrow text{ the distribution of } X/theta text{ is independent of } theta$$

If $X$ ~ $LogNormal(theta, 1)~, theta in (0, infty)$ (We restrict the parameter space) then $X/theta $~ $LogNormal(theta + ln(1/theta), 1)$. With this definition $theta$ is not a scale parameter.

I was wondering which of these two is the correct definition of a scale parameter? Or it depends on the book and the author?