I have seen many different definitions of what it means for a curve to be "smooth". In this question, for instance, a curve $\gamma \colon [a,b] \longrightarrow \mathbb{R^n}$ is defined to be smooth if all derivatives exist and are continuous. This seems reasonable and in-line with what it means for any manifold to be smooth.
See, however, Edwards Calculus of Several Variables where a smooth curve is defined (actually, he uses the word "path"):A curve $\gamma \colon [a,b] \longrightarrow \mathbb{R^n}$ is said to be smooth if the derivative $\gamma^{\prime}(t)$ exists, is continuous and if $\gamma^{\prime}(t) \neq 0 \;\; \forall t \in [a,b]$.
Ignoring the fact that Edwards is only concerned with $C^{1}$ curves as opposed to $C^{\infty}$ curves, that this definition requires the derivative to be nonzero obviously makes it different from the first definition.
Now, consider yet another definition, this time from Wade's Introduction to Analysis (3rd Edition): A subset $C$ of $\mathbb{R}^m$ is called a $C^p$ curve in $\mathbb{R}^m$ if and only if there is a nondegenerate interval $I$ and a $C^p$ function $\gamma \colon I \longrightarrow \mathbb{R}^m$ that is injective on the interior of $I$ and $C = \gamma(I)$
Ignoring the fact that Wade is defining a curve as a set of points instead of its parametrization, this definition also differs from the first two in that $\gamma$ is required to be injective.
So, finally, here is my question: Is there a definition, or set of definitions, that could bring some consistency to this terminology? Perhaps there's a special name for a "curve" that is "injective" in addition to being "smooth"? Or, maybe there's a special name for a curve that never evaluates to the $0$-vector anywhere on it's domain? It seems to me that these are different attributes and should probably have distinctive nomenclature.