How can we relate the two informal definitions of the limit with each other?

The second informal definition is more precise. Here's the problem with the first one: consider $f(x) = x^2$, and take $x \rightarrow 0$. Now I claim that the limit is $-1$:

As $x$ gets closer and closer to $0$, $f(x)$ gets closer and closer to $-1$.

This statement is true, but the actual limit is certainly not $-1$.

The problem with the first definition is that it's so informal that it's missing the critical part of the concept, which is that I should be able to make it as close as I want to $-1$ by choosing values of $x$ close to $0$ (which I can't in this case). That precision is what the second definition supplies.

The first informal definition suggests motion: as $x$ somehow "approaches" or "gets closer to" $a$, the value $f(x)$ approaches $L$.

The second informal definition is actually the formal definition, written in English. Mathematicians designed it to capture the idea of "approaches" without the need for vague terms like "closer and closer to".

The English phrase "arbitrarily close to $L$" is written more formally as "closer than $\epsilon$ for any particular $\epsilon$ you may have in mind." The "arbitrary" says "for any possible $\epsilon$."

The "sufficiently close to $a$" is written more formally as "you can find a $\delta$ (which will usually depend on $\epsilon$) that will guarantee $f(x)$ within $\epsilon$ of $L$ as long as $x$ is within $\delta$ of $a$." The $\delta$ is sufficient (i.e. good enough) to guarantee the $\epsilon$ closeness of $f(x)$ to $L$.