I have a very strong intuition about the equivalency of the following definitions for Cauchy sequences:
$$\forall \varepsilon > 0, \exists M \in \mathbb{N} : k, l \geq M \implies d(x_k, x_l) < \varepsilon $$
$$ \exists N \in \mathbb{N} : n \geq N \implies \exists \varepsilon > 0, x : x_n \in B(x, \varepsilon) $$
The way I see it, you can potentially (and temporarily!) 'fix' either $k$ or $l$ in the first definition and use it as a center for the ball of the second. In this way, you could suppose that the ball actually has a diameter of$\varepsilon$, with 'nearby' points $x_k$ and $x_l$.
In the second definition, saying that $x_n \in B(x, \varepsilon)$ is equivalent to saying the distance from $x$ to $x_n$ is less than $\varepsilon$, which would relate it back to the first definition.
I know beyond a doubt that these definitions must be equivalent, but I can't seem to formalize my intuitions in a way that is irrefutable (e.g. neither $x_k$ nor $x_l$ need be in every $B(x, \varepsilon)$ for all $\varepsilon$, thus the need for 'temporarily').
Is there a better way of saying what I'm thinking?
They are actually not equivalent. Your second definition simply says that the sequence is (eventually) bounded. The first implies the second, but the second does not imply the first.
For example, consider the sequence $x_n:=(-1)^n$. This is certainly not Cauchy--try it for any $\epsilon\in(0,2).$ However, for any $N$ and all $n\ge N$, it is certainly the case that $x_n\in B(0,2)$.
The big difference, here, is that the definition of Cauchy allows us to take $\epsilon$ to be any positive number, and still see that, for big enough $n$, all the $x_n$ are within $\epsilon$ of each other. In your definition, we're only guaranteed that all $x_n$ will (eventually) lie within some $\epsilon$-neighborhood of some point, but we have no control over how big this neighborhood is allowed to be.