Okay, I keep reading the definitions over and over, but I don't see the difference between the two ; apparently spaces that deformation retract onto a point are contractible, but the opposite is not necessarily true.
Definition of contractible : A space $X$ is said to be contractible if it has the homotopy-type of a point, i.e. there is a map $f : X \to \{p\}$ and a map $g : \{p\} \to X$ such that $fg \simeq \mathbb 1_p$ and $gf \simeq \mathbb 1_X$. Since $fg = \mathbb 1_p$, it suffices to show that there is some point in $X$ such that the identity map $\mathbb 1_X$ is homotopic to a constant map at some point $g(p) \in X$.
Definition of deformation retract onto a point : there exists a map $f : X \times I \to X$ such that $f(x,0) = \mathbb 1_X$ and $f(x,1) = p$ for some fixed point $p \in X$ and $p$ does not depend on $x$.
I believe an homotopy from the identity map to the constant map is precisely a deformation retract, so I don't see the difference. Or maybe I got the definitions wrong. Does anybody see it? I am quite stuck, since one exercise asks to show that some weird space is contractible but does not deformation retract onto a point.