I have used the cofiber sequence many times. I decided to try and derive it myself since I had managed to forget it.
Let $A \hookrightarrow X$ be a cofibration. Consider the projection $X \to X/A$. The 'next term' in the cofibration sequence is the cofiber of this map when I turn $X \xrightarrow{q} X/A$ into a cofibration. The standard way of doing this is substituting for the inclusion of $X \hookrightarrow M_q$. The cofiber of this map is $M_q/X:=\frac{(X \times I)/((x,1) \sim (\overline{x},1))}{X \times 0}=Cone(X/A)$.
This is not the standard correct answer I get by substituting the map $X \to X/A$ by $X \to Cone(A \hookrightarrow X)$(the cofiber of this map is the suspension of $A$).
What is the flaw in my first argument?