@Thomas is right -- in my book, curves are always treated as parametrized curves, that is, smooth (or sometimes piecewise smooth) maps from an interval into the manifold. The $t$ in the notation $D_t\gamma$ refers to the independent variable, i.e., the standard coordinate on $\mathbb R$. It's parallel to the notations $d/dt$ or $\partial/\partial x$ for ordinary derivatives. (Some authors use $D\gamma/dt$ for the covariant derivative along a curve, but to my eye that looks ugly and cumbersome.) All of these notations are illogical, strictly speaking, because the name of the independent variable is not an intrinsic property of a function. But they're too convenient to abandon just because they don't happen to be logically consistent!
The situation in which the $D_t$ notation really shows its power is when considering a parametrized family of curves (see Chapter 6 in my book). Given a smooth map $\Gamma\colon (-\epsilon,\epsilon)\times[a,b]\to M$, we can consider the curves $s\mapsto \Gamma(s,t)$ for fixed $t$, or the curves $t\mapsto \Gamma(s,t)$ for fixed $s$, and denote the covariant derivatives along these curves by $D_s$ and $D_t$, respectively. Any other notation quickly becomes unwieldy.
I'm going to be working on a second edition of my Riemannian Manifolds book during the next couple of years, so I welcome suggestions.
Jack Lee