Let $f':M \to A$ and $g':M \to B$ be $R$-module homomorphisms, and let $$Y = A \oplus B \Big / \{(f'(m),-g'(m)\;|\;m \in M\}$$ be a quotient. How will the maps $\pi_1:A \to Y$ and $\pi_2:B \to Y$ in the diagram below, be defined explicitly?
$\require{AMScd}$ \begin{CD} M @>g'>> B\\ @V f' V V @VV \pi_2 V\\ A @>>\pi_1> Y \end{CD}
Edit: I believe $$\pi_1(f'(m)) = (f'(m),0)$$ and $$\pi_2(g'(m)) = (0,g'(m)).$$
Write $p : A \oplus B \to A \oplus B \big / \{ (f'm, -g'm) \mid m \in M \}$ for the usual quotient map, sending $(a,b)$ to the coset $(a,b) + \{ (f'm, -g'm) \mid m \in M \}$.
Then $\pi_1 : A \to Y$ is $\pi_1(a) = p((a,0_B))$ and $\pi_2 : B \to Y$ is $\pi_2(b) = p((0_A,b))$. If you prefer, these are the composites of the inclusion maps $\iota_A : A \to A \oplus B$ and $\iota_B : B \to A \oplus B$ with the quotient map $p : A \oplus B \to A \oplus B \big / \{ (f'm, -g'm) \mid m \in M \}$.
Moreover, you can check that this agrees with your intuition that $\pi_1(f'm)$ is sent to the coset of $(f'm,0)$ in the quotient (and similarly for $\pi_2(g'm)$).
I hope this helps ^_^