Motivation of irrationality measure

I have a question about the irrationality of $e$:

In proving the irrationality of $e$, one can prove the irrationality of $e^{-1}$ by using the series So the series for $e^{-1}$ is so that or If $e^{-1}$ were rational then we would have a difference of two integers which is not an integer (i.e. between $0$ and $\frac{1}{2}$).

Question: Is this "irrationality of $e^{-1}$ by alternating series" proof what motivated the definition of irrationality measure? It seems that this was born out of using alternating series.

Does the same method work for $\pi$ ?

The motivation for the definition of irrationality measure came from results in diophantine approximation. E.g., Dirichlet's Theorem states that for every real irrational $$xx$$ there are infinitely many integer pairs $$p,qp,q$$ with $$|x−(p/q)|. If $$xx$$ is a real quadratic irrational, then there's a positive constant $$CC$$ such that for every integer pair $$p,qp,q$$ we have $$|x−(p/q)|>Cq−2|x-(p/q)|>Cq^{-2}$$. Liouville's Theorem was mentioned in the first comment. These can all be stated in terms of irrationality measure, and form an obvious motivation for that concept.