The definition of Liouville number $l$ is, for every positive integer $n$, there exist integers $p,q$, such that $$\left\vert l-\frac{p}q\right\vert<\frac1{q^n}$$ is satisfied.
However, in an accepted and upvoted answer on MSE, it seems that one can choose $p,q$ arbitrarily. So, the definition of Liouville number should have ’for any’ replacing ’there exist’.
What confused me more is that the Liouville approximation theorem states that:
For any algebraic number $x$ of degree $n>2$, a rational approximation $\frac{p}q$ to $x$ must satisfy $$\left\vert x-\frac{p}q\right\vert >\frac1{q^n} $$
This makes me guess that the inequality for Liouville numbers is satisfied for any $(p,q)$.
Which is correct and which is wrong?
The first definition you gave is correct. I recommend you re-read what you linked. He said that pi is not a liouville number. The negation of there exists is for all.