Clarification about the definition of Direct limit of modules.

Question

Let $A$ be a ring, $(M_i)_{i \in I}$ the family of modules that will bring the direct limit, and $f_{ij} : M_i \to M_j$ the transition maps. I have two definition of direct limit of $M_i$:

1. Wikipedia (https://en.wikipedia.org/wiki/Direct_limit)
2. Exercise 14, pag.33 of the book by Atiyah and MacDonald, equal to what you can find on this link (https://ysharifi.wordpress.com/2011/01/17/direct-limit-of-modules-1/)

There is a slightly difference between them regarding the quotient operation. After having taken the direct sum of all the modules...

1. Wikipedia just make the quotient by the equivalent relation $x_i\sim\, f_{ik}(x_i)$;

2. On the book (or in the link) we define as N the submodule generated by the elements of the form $x_i - f_{ik}(x_i)$ (identifying each element of $M_i$ in the direct sum), and the we make the quotient with that submodule.

At first look, they could appear equivalent, but...I am not so sure, and I would be grateful if someone could help me in confirm the difference that I found after some computation. According to what I manage do understand..

1. the equivalent relation $. x_i\sim\, f_{ik}(x_i)$ is not compatible with the sum operation of a module;
2. Although this difficulty, we want to introduce a relation for which $. x_i\sim\, f_{ik}(x_i)$:
3. We make the submodule as indicate. Notice that with the quotient in that way, actually "$ x_i\sim\, f_{ik}(x_i)$" but it is not an iff, in the sense that if $[x_i] = [x_j]$ then $x_i - x_j \in N$ and it not suffice to conclude just $x_j = f_{ij}(x_i)$.

Is it correct? Thank you very much in advance.

Cheers