For brevity I'm making the following assumption: I'm only talking about regular curves on $\left[a,b\right]$ with values in $\mathbb{R}^{n}$, and line integrals of scalar fields.
[Since there are a lot of questions at the end and you have to dig through the text below to make sense of them, I'm willing to offer 150 bounty points for a complete and thorough answer by a knowledgeable person in either differential geometry or vector analysis (or related fields) of all the questions.
There are three common ways to define curves:
Here they are defined as mappings $\gamma:\left[a,b\right]\rightarrow\mathbb{R}^{n}$.
Here a curve is used as a subset $C$ of $\mathbb{R}^{n}$ that's the image of a (regular) $\gamma:\left[a,b\right]\rightarrow\mathbb{R}^{n}$- which clearly doesn't work with the concept from 1. (although Wikipedia has linked to that - has no one spotted this so far ?)
Here curves are equivalence classes of mappings $\left[a,b\right]\rightarrow\mathbb{R}^{n}$ (which are equivalent if they are obtained from reparametrisations of eachother).
Now these definitions relate in different ways to the concept of a line integral over scalar field $f$ along $\gamma$.
If I take 1. as my definition, I can define $$ \int_{\gamma}fd s:=\int_{a}^{b}f\left(\gamma\left(t\right)\right)\left\Vert \dot{\gamma\left(t\right)}\right\Vert d t $$ without mathematical problems, but I have the "psychological" problem that I would like my line integral to not be dependent on all the information $\gamma$ contains (since, for example, different reparametrisations of $\gamma$ give me the same $\int_{\gamma}f$) - of course I can show that this holds in a separate theorem, but this just seems ugly).
If I take 2. as my definition, I can define $$ \int_{C}fds:=\int_{a}^{b}f\left(\gamma\left(t\right)\right)\left\Vert \dot{\gamma\left(t\right)}\right\Vert d t $$ where $\gamma$ is any parametrisation $\gamma:\left[a,b\right]\rightarrow\mathbb{R}^{n}$ that has image $C$. This has mathematical problems: There are (regular) curves that have the same image , but aren't equivalent, so have different lengths. For example $$ t\mapsto\left(\begin{array}{c} \cos t\\ \sin t \end{array}\right)\ \text{and}\ t\mapsto\left(\begin{array}{c} \cos2t\\ \sin2t \end{array}\right) $$ where $t\in\left[0,2\pi\right]$. So $\int_{C}f s$ isn't well defined since it depends on the choice of the parametrisation of $C$. But from a "psychological" perspective I like this the most, since it only has a geometric content (since $C\subseteq\mathbb{R}^{n})$ and not a dynamic one (I don't know anything about the "speed" with which $C$ is traced) and my personal view is, that line integral (or arc lengths, since I could have discussed this issue in the same matter with arc lengths instead of line integrals) should be purely geometrical.
If I take 3. as my definition, I can define
$$
\int_{\hat{\gamma}}fd s:=\int_{a}^{b}f\left(\gamma\left(t\right)\right)\left\Vert \dot{\gamma\left(t\right)}\right\Vert d t
$$
where $\hat{\gamma}$ is the equivalance class of $\gamma:\left[a,b\right]\rightarrow\mathbb{R}^{n}$.
This seems to me to be a compromise between 1. and 2.: The line integral
is (from the start) independent of reparametrisations - but it isn't
purely geometric, as the definitions from 2.
But other weird issues arise in this case: Since a curve is a set of mappings (which makes up the equivalence class) I loose a comfortable way of speaking about curves by the following subtle point: I can't say anymore that a smooth curve is also a continuous curve, since for example the equivalence class of the identity on $[0,1]$ (taken as a curve in $\mathbb{R}$), viewed as a smooth curve (i.e. as the set $\{f\mid f:[0,1] \rightarrow \mathbb{R} \text{ is smooth and can be reparametrised to be the identity}\}$) does not contain $$t\mapsto\begin{cases}
2t, & 0\leq t\leq\frac{1}{2}\\
1, & \frac{1}{2}<t\leq1
\end{cases}$$which is is in the equivalence class of the identity, viewed as a continuous curve (i.e. as the set $\{f\mid f:[0,1] \rightarrow \mathbb{R} \text{ is continuous and can be reparametrised to be the identity}\}$) .
Questions:
A. Is there a standart definition of what a curve (and thus a line integral) is ? If there isn't a definition that's valid for the whole of mathematics, is the definition at least separately standarised in subfields (like differential geometric, vector analysis etc.) ?
B. Is my view that line integrals and arc lengths should only depend on a purely geometric object of a curve "correct" ? (You may understand what you wish by "correct".)
C. Could I perhaps save the definition of $\int_{C}f ds$ from 2., by modifying its definition so that it says $$ \int_{C}fds:=\int_{a}^{b}f\left(\gamma\left(t\right)\right)\left\Vert \dot{\gamma\left(t\right)}\right\Vert d t $$only for those $\gamma$ that are injective on $\left(a,b\right)$ ? (In this case I would need a proposition, that for every $\gamma:\left[a,b\right]\rightarrow\mathbb{R}^{n}$ there exists an injective $\gamma':\left[a,b\right]\rightarrow\mathbb{R}^{n}$, such that $\gamma$ and $\gamma'$ have the same image. Does such a proposition exist ?)
Would I exclude important physical phenomena by this alternative definition of 2.?
D. I've know that there also a fourth definition if a curve, namely as a topological space locally homeomorphic to a line. How does this definition reduce to each of the three definitions above (as Wikipedia says at the beginning of the article http://en.wikipedia.org/wiki/Curve about curves) and how do I define a line integral (or arc length) by this definition ?
Note: I've already read this in case you wanted to direct me there.
This is a nice question IMHO, and my take is as follows (not a full answer maybe and not bounty worth btw):
The image definition is certainly ill-posed, as your example illustrates. Possible fix: $\forall p\in C$, define $t_p = \min\{t: \gamma(t) = p\}$, and redefine $\gamma = \gamma|_{D}$ where $D = \cup_{p\in C}t_{p}$. BTW Wikipedia says the following in the entry of Curve:
Mathematically, I prefer to define in the first way, i.e., as a mapping to avoid ambiguity so that different parametrizations lead to the same value for the line integral. A definition restricting an oriented curve to have a designated parameterisation is good for me too. "Psychologically", I prefer the physical way, i.e., view $f = F\cdot \gamma'/\|\gamma'\|$, provided there exists such field $F$. Then the scalar integral is the same as the work done by the field $F$ when a unit mass point particle traverses the whole curve $\gamma$ from $\gamma(a)$ to $\gamma(b)$. In your first example, if using the second parametrization, the particle rotates the circle twice, and we have twice the work done.
I agree with you on that "the line integrals and arc lengths should only depend on a purely geometric object of a curve", so it is "subjectively correct" for myself :) The reason I prefer the physical way, is that, we can then associate the line integral with the bilinear dual pair between forms and chains. Bilinearity is in that $$ \langle \gamma, a_1 \omega_1 + a_2\omega_2\rangle = \int_{\gamma} (a_1 \omega_1 + a_2\omega_2) = a_1 \int_{\gamma}\omega_1 + a_2\int_{\gamma}\omega_2, $$ and $$ \langle c_1\gamma_1 + c_2 \gamma_2, \omega\rangle = \int_{c_1\gamma_1 + c_2 \gamma_2} \omega = c_1 \int_{\gamma_1}\omega + c_2\int_{\gamma_2}\omega. $$ The linearity in $\gamma$ essentially rules out the ambiguity in definition 2. For in this view, the "curve" is viewed as an element in the 1-chain vector space, which includes the piecewise smooth curves being continuous not smooth. By your example in the third definition, reparametrization seems working fine "psychologically" on for curves such that $\gamma'(t)$ not vanishing for all $t$, for now smooth curves are continuous curves when the equivalence restricts on $\gamma$ of which $\gamma'(t)\neq 0$.