Understanding the definition of inert functions in Kiral–Petrow–Young

I am reading the paper Oscillatory Integrals with Uniformity in Parameters by Kiral, Petrow, and Young, and I am having trouble understanding the notion of an inert function introduced in Definition $2.1$.

These are my confusions Since $X=X_T \in [1,\infty]$.Then if we consider a family of function which is $1-inert$, then it is $c-inert$ for any constant $c \in [1,\infty]$ because of the inequality that the LHS is finite. It does not say anything else. I am unable to differentiate what is the difference between $1-$inert and $2-$inert function.

However let us say that, a function is $x-$inert, then we do have something concrete here, a function can be $x-$inert but I am not sure about where this $x$ would come from inside the family.

Could it be something like, we have a parameter $x$ which is used to define a specific family of functions $\{f:\mathbb{R} \to \mathbb{R}\}$ and on basis of that, we use a variable $x$ here. More accurately, $x$ is another variable, so while any $1-$ inert function is $c-$inert, but we cannot say that $x-$inert function is $\sqrt{x}-$inert.

Reference: E. M. Kiral, I. Petrow, M. P. Young, Oscillatory Integrals with Uniformity in Parameters.