What is the difference between a power law exponent and the Pareto exponent?

I use the poweRlaw package in R to fit a power law to my data. I am trying to figure out what is the value of the Pareto exponent.

Assume the probability mass function is defined by:

$$p(x) = \frac{\alpha-1}{x_{min}} \left(\frac{x}{x_{min}} \right)^{-\alpha}$$

and the complementary cumulative density function is defined by:

$$P(x) = \int_x ^\infty p(x’) dx’ = \left(\frac{x}{x_{min}}\right)^{-\alpha + 1}$$

Is the Pareto exponent $$\alpha$$ or $$- \alpha + 1$$ or $$\alpha - 1$$? In most literature, the CCDF is used to describe the income/wealth distribution, and $$-(\alpha-1)$$ is the slope of the CCDF on a log-log plot, so $$\alpha-1$$ seems the most intuitive. I'm pretty sure the R library poweRlaw returns $$\alpha$$ as defined above. I am using Newman as a reference.

A random variable $$X$$ has a Pareto distribution with Pareto exponent $$\theta$$ if $$\text{P}(X>x)=\begin{cases} \left(\frac{x}{x_m}\right)^\theta \quad \text{ if } x\geq x_{min}\\ \ \ 1 \quad \quad \quad \text{ if } x
In this case, the Pareto exponent is $$\theta = \alpha - 1$$. Remember that $$P(X>x)=\int_x^\infty p(x')\mathrm{d}x'$$.