Properly Defining a Smooth Curve 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.
 A: Yes, there is a set of definitions that can bring consistency to the terminology.
A curve $\gamma\colon I \to \mathbb{R}^n$ is smooth iff it is $C^\infty$ (or $C^p$ for some authors).
A curve $\gamma\colon I \to \mathbb{R}^n$ is an immersion iff it is $C^\infty$ and $\gamma'(t) \neq 0$.
A curve $\gamma\colon I \to \mathbb{R}^n$ is simple iff $\gamma$ is injective on the interior of $I$.
(The reason for requiring injectivity on the interior, rather than on the whole interval, is so that we may legitimately speak of "simple closed curves."  That is a simple closed curve $\gamma\colon [a,b] \to \mathbb{R}^n$ is a map that is injective on $[a,b)$ and has $\gamma(a) = \gamma(b)$.)
Granted, these definitions aren't 100% universal, but I think they're fairly standard.  The term "regular" is sometimes used as a synonym for "immersion," but then again, "regular" can mean other things in other contexts, too (which is why I prefer to avoid the term altogether if I can help it).
And as you point out, some authors refer to a curve as the image of such a map, rather than the map itself.
A: I stumbled upon this old question and gave an pointer in another thread https://math.stackexchange.com/a/4167512/799131
But maybe it's good to just copy it here:
I'd like to add something: there is a difference of perspective on smoothness depending whether you look at the geometric object or its parametrization.
Look at the standard example: the real cusp. It is a curve in the real plane parametrized $f:t\to (t^2,t^3)$. Of course, the mapping $f$ is smooth (of any order), and the graph of $f$ is a smooth manifold in $\mathbb{R}^3$, but its  image is singular: it is the zero set $x^3=y^2$. It is "worse than a corner"!
So you need to be always clear what you want: do you need only differentiability of the parametrization or do you want the image to be a differentiable manifold (typically in such a case you would assume that the derivative of $f$ does not vanish).
