The definition of homotopy in algebraic topology

In this post, let $I=[0,1]$.

Something about the definition of homotopy in algebraic topology (and in particular in the study of the fundamental group) always puzzled me. Most books on the fundamental group often begin with the basic notion of a homotopy of curves (or more generally, continuous functions between topological spaces) and describe it intuitively as "a continuous deformation of one curve into another". They often supplement this statement with some nice picture, like this one in Wikipedia. When I was taught algebraic topology, I too had heard a motivating explanation as above and was shown a picture of this sort. From this I could already guess what a (supposedly) natural formal definition would be. I expected it to look something like this:

Let $X$ be a topological space and let
$f,g : I to X$ be two curves in $X$.
Then a homotopy between $f$ and $g$
is a family of curves $h_t: I to X$
indexed by $t in I$ (the "time"
parameter) such that $h_0 = f$, $h_1 =
> g$ and the function $t mapsto f_t$ is
continuous from $I$ to $C(I,X)$ (the
space of curves in $X$ with domain
$I$, equipped with some suitable
topology).

However, the definition given (which is used in every book on algebraic topology which I sampled) is similar, but not quite what I thought. It is defined as a continuous function $H: I times I to X$ such that $H(s,0)=f(s)$ and $H(s,1)=g(s)$ for all $s in I$.

This actually quite surprised me, for several reasons. First, the intuitive definition of a homotopy as a "continuous deformation" contains no mention of points in the space $X$ - it gives the feeling that it is the paths that matter, not the points of the underlying space (though obviously one needs the space in question to define the space of paths $C(I,X)$). However, the above definition, while formally almost equivalent to the definition I thought of (up to a definition of a "good" topology on $C(I,X)$), makes the underlying space $X$ quite explicit, it appearing explicitly in the range of the homotopy.

Moreover, many of the properties related to homotopies, the fundamental group and covering spaces can be expressed using the vocabulary of category theory, using universal properties. Now, from a categorical-theoretic point of view, wouldn't one want to suppress the role of the underlying space as much as one can (in favor of its maps and morphisms)?

Additionally, the definition of homotopy (as used) seems notationally inconvenient to me, in that it is less clear which of the two variables is the time parameter (each mathematician has his own preference, it seems). Also, the definition of many specific homotopies looks needlessly complicated in this notation, IMO. For instance, if $f,g$ are two curves in $mathbb{R}^n$ then they are homotopic, and one can write the obvious homotopy either as $H(s,t)=tf(s)+(1-t)g(s)$ or as $h_t = tf+(1-t)g$. Maybe that's just me, but the second notation seems much more natural and easier to understand than the first one. Formulae of this sort appear frequently in the study of the fundamental group of various spaces (and in the verification that the fundamental group is indeed a group), and using the $H(s,t)$ notation makes these formulae much more cumbersome, in my opinion.

So, to sum up, I have two questions:

1) For a topological space $X$, can
$C(I,X)$ be (naturally) topologized so
that "my" definition of homotopy (see above) and
the usual definition coincide (by
setting $h_t (x) = H(x,t)$)?

2) If so, why isn't such a definition
preferred? See my arguments above.