On the Leibniz's test for alternating series 
Given a monotonically decreasing sequence $a_n$ which converges to
  $0$, then the alternating series
$$\sum_{n=1}^{\infty} (-1)^n a_n$$
converges.

This is the statement of the Leibniz's test.
My question is very simple. Suppose that $a_n$ is monotonically increasing and it's limit is $0$. Then, I can say that:
$$\sum_{n=1}^{\infty} (-1)^n (-a_n)$$
converges using the Leibniz's test. But, at this point I can also conclude that 
$$\sum_{n=1}^{\infty} (-1)^n a_n = -\sum_{n=1}^{\infty} (-1)^n (-a_n)$$
converges too.
Should the Leibiniz's test be restated as follows?

Given a monotonic sequence $a_n$ (i.e. no matter if it is increasing or decreasing) which converges to
  $0$, then the alternating series
$$\sum_{n=1}^{\infty} (-1)^n a_n$$
converges.

 A: Yes, you are completely right. The theorem is more often stated in its simpler form, and this is not very harmful, since you verified that it's easy to derive the more complete form from the simpler. But indeed, your new version of the theorem is stronger and correct.
EDIT: I realized that I didn't exactly answer your question: Should the Leibniz test be restated? I think that this is a matter of opinion. In my opinion, if you are writing a text in which the stronger version will make it easier to derive most of your results, you should state it so, but otherwise, I think that the simpler form of the theorem is preferred. (Yes, in my conception of "simple", a monotonically decreasing sequence converging to zero is simpler than a monotonic sequence converging to zero. But again, this is a matter of opinion).
A: It is a question of personal opinion as mentioned by fonini.
For myself, I find it more pedagogical to  provide the statement of a theorem with many conditions to bring up the heart of the theorem to the reader (student). 
Next, some illustration examples to show how this theorem works in practice. Then, I mention as a remark that such and such hypothesis can be relaxed or weakened a little bit. If possible I give some example(s) of these weakened versions. 
For the example of Leibniz test, the general version mentioned by OP can be given as a second version, or in a remark. 
Also, we can give as a second version that the decreasing of the sequence $a_n$ can start from  $n=1$ but it is not necessary, it can also start from  $n=n_0$ for any  $n_0 \geq 1$.
If a particular student in the class notice that from the general version one can extend the result to some particular case, then he deserves congratulations!
