Difference between Deformation Retraction and Retraction

The difference between a retraction and a deformation retraction does have to do with the "notion of time" as you suggest.

Here's a strong difference between the two:

1) For any $x_0 \in X$, $\{x_0\} \subset X$ has a retract. Choose $r : X \to \{x_0\}$ to be the unique map to the one-point set. Then, certainly, $r(x_0) = x_0$.

2) However, $\{x_0\} \subset X$ only has a deformation retraction if $X$ is contractible. To see, why, notice there has to be a family of maps $f_t : X \to X$ such that $f_0(x) = x$, $f_1(x) = x_0$, and $f_{t}(x_0) = x_0$ for every $t$. This gives a homotopy from $id_X$ to the constant map at $x_0$, which makes $X$ contractible.

In fact, showing a deformation retract from $X$ onto a subspace $A$ always exhibits that $A$ and $X$ are homotopy equivalent, whereas $A$ being a retract of $X$ is weaker. (But often, still useful! Two spaces being homotopy equivalent is very strong indeed!)

As you noted the two notion are different, deformation retraction being a continuous family of continuous function, i.e. an homotopy, while a retraction being just a continuous function.

A retraction is just a map that sends all the point of $X$ in $A$ fixing the points of $A$.

A deformation retraction as the opposite is a family of mappings that fix the points of $A$, but that's more: we require that the family is continuous that means that we want that the induced map $X \times I \to X$ sending every pair $(x,t) \in X \times I$ to $f_t(x)$ is a continuous function.

Anyway there's also another way to see deformation retraction and more generally homotopies. For every space $Y$ we can consider the set $Y^I=\mathbf{Top}(I,Y)$ the set of continuous paths in $Y$ and topologizes this set with the compact-open topology.

Since the space $I$ is locally compact a general theorem tells that there's a bijection $$\mathbf{Top}(X \times I , Y) \cong \mathbf{Top}(X,Y^I)$$ sending every map $F \colon X \times I \to Y$ in the map $\bar F \colon X \to Y^I$ that to every $x \in X$ associates the continuous function $\bar F(x) \colon I \to Y$ such that for $t \in Y$ $\bar F(x)(t)=F(x,t)$ (this bijection in natural both in $X$ and $Y$).

Because of this bijection we can define an homotopy to be just a continuous function in $\mathbf{Top}(X,Y^I)$.

If we adopt this this point of view, of homotopies as continuous mapping in path spaces, a deformation retraction is just a mapping that associate to every point $x \in X$ a path, starting at the point $x$ and ending at some point of $A$. Each path corresponding to a point is the trajectory that the point follows during the deformation of $X$ to $A$.

Anyway a deformation retraction of $X$ to $A$ is not simply an homotopy: it is also an homotopy relative to the subspace $A$ between the identity and a map $r \colon X \to X$ such that $r(X) \subseteq A$. By the requirement that $r$ is homotopic to $1_X$ via an homotopy relative to $A$ it follows that $r$ must be a retraction: by definition an homotopy relative to $A$ sends every point of $A$ into the constant path which connect $a$ to $r(a)$, which must be equal.

This means that if $A$ is a deformation retract of $X$ then $A$ is also a retract, anyway the converse doesn't hold in general: a counterexample consider the map that send $S^1$ (the circle) to a point, this is a retraction, but there's no deformation retraction of $S^1$ to a point (to prove this one need to make a little work and build some invariant like the $\pi_1$ which is in the next chapter of the book).

Hope this helps understanding the ideas and the differences of these two concepts.