Definition of Cauchy Sequence

Here is what I think. First of all, as @Surb pointed out, $\Rightarrow$ is shorter than $\Leftarrow$ (and in fact is pretty short in its own right). It can then be seen that your second definition is an almost immediate corollary of the original, whereas the first definition is definitely not as immediately deducible from the one you found. So in practice, if you wanted to use your definition instead of the original, you could easily just derive it from the original. On the other hand, suppose you had your definition and in some situation you found that the original would be more useful. It would take a lot more work to derive it from your definition, so that's a little bit inconvenient. But really this is just a question of convenience.

Also, and this is more a statement about intuition than anything, I think the original definition expresses a particular intuitive point more clearly than your second one does. The original basically says a sequence is Cauchy if the terms become arbitrarily close to one another. Your definition essentially conveys the same point, but if you think about it, it isn't as obvious from that definition. They are, as you've shown, equivalent so you could obviously use whichever you want.

In any context, the "best definition" of a property should be as clear, natural, and intuitive as possible. That's my belief.

Sometimes this comes at a cost. For example, I favor the following definition of linear independence in a vector space: The vectors $v_1,\dots,v_n$ are linearly independent if no $v_k$ is a linear combination of the other $v_j.$ This definition seems to me to get at the heart of the matter more naturally than the usual definition. However, there is a cost to this: the usual definition turns out to be easier to apply most of the time. Fine. In my book, it's best to be kind to your audience and start with the most natural definition, then prove it's the same as the one given in umpteen texts. Then you're off to the races, and the audience, without knowing it, is better off at the expense of an extra "dt" in the time it might take.

In the case of Cauchy sequences, the "usual" definition has one more "variable" than the definition proposed. It is claimed that therefore the proposed definition is simpler. But I don't think it is simpler, nor better, nor more natural. The usual definition tells me that after some point, all terms of the sequence are "close to each other." And I think this is the intuition we're trying to get at. The proposed definition tells me that after the index $N,$ all terms are "close to $x_N$". But this $x_N$ is a moving target! Thus there is a hidden variable in the alternate definition. Why try to hide it? Let's bring it out into the light and deal with it instead of brushing it under the rug like a politician.

You just rediscovered Cantor's original definition. A good discussion is here. Its wording is slightly different from yours: $$\forall m \in \mathbb{N}\ \lim_{n\rightarrow \infty} \left (x_{n+m}-x_n \right ) = 0 $$

And I think that this definition is more intuitive.

You can give the following justification. You can start with the following exercise:

Theorem 1 If the sequence $\{x_n\}$ is convergent then $\displaystyle{\lim_{n\rightarrow \infty}} \left (x_{n+1}-x_n \right ) = 0 $.

Then you ask the question: is the converse true? Of course not. But you can suggest another exercise
Theorem 2 If the sequence $\{x_n\}$ is convergent then $\displaystyle{\lim_{n\rightarrow \infty}} \left (x_{n+2}-x_n \right ) = 0 $.

Is the converse true here? Again the answer is negative. Does the conclusion of Theorem 1 imply the conclusion of Theorem 2 or other way around? The answer is still negative. Do the conclusions of Theorem 1 and Theorem 2 together imply the convergence? Additional negative answer.

In a similar way you can state Theorem 3, ..., Theorem $m$. Now you can ask: if you take the conclusions of Theorem 1, ..., Theorem $m$ as assumptions, does imply the convergence of the sequence $\{x_n\}$. You can check this for $m=3, 4$ and collect additional negative results. (All these trials can supply a lot of meaningful drill exercises for proving convergence or finding counterexamples)

After all these preparations, as a last resort you can propose: If we assume the conclusions of Theorem $m$ for all $m$, i.e.

if you asssume that $$\displaystyle{\lim_{n\rightarrow \infty}} \left (x_{n+m}-x_n \right ) = 0 $$ for all $m \in \mathbb{N}$ does this imply convergence?

And still you cannot say yes! In the example given above we have
$$\lim\limits_{n\rightarrow \infty} \left (x_{n+m}-x_n \right ) = 0 $$ for each $m\in \mathbb{N}$.

The difference between this condition and Cantor's definition is that the convergence in Cantor's definition is uniform in $m$ (the $\varepsilon$ does not depend on $m$).