In the context of evaluating the line integral of a vector field $\vec{F}:\vec{x}\in\mathbb{R}^n\mapsto \vec{F}(\vec{x})\in\mathbb{R}^n$ along a regular parameterized curve $\gamma:t\in{I}\subseteq\mathbb{R}\mapsto \vec{x}(t)\in\mathbb{R}^n$, we define $$\int_{\gamma}\vec{F}\cdot d\vec{x} = \int_{I}\vec{F} \left(\vec{x}(t)\right)\cdot \frac{d\vec{x}}{dt}(t)dt$$
I have learned that this definition ensures its invariance under an orientation-preserving diffeomorphism $\tau:t\in I \mapsto \tau(t)\in I'$ since the chain rule provides that $$\frac{d\vec{x}}{d\tau} (t(\tau)) = \frac{d\vec{x}}{dt} (t(\tau)) \frac{dt}{d\tau} (\tau)$$ and the change in measure introduces a factor $\lvert{\frac{dt}{d\tau}(\tau)}\rvert$ such that $$\int_{\gamma} \vec{F} \cdot d\vec{x} = \int_{I'}\vec{F}(\vec{x}(t(\tau))) \cdot \left[\frac{\frac{d\vec{x}}{d\tau}(t(\tau))} {\frac{dt}{d\tau}(\tau)}\right] \lvert \frac{dt}{d\tau}(\tau)\rvert d\tau = \int_{I'}\vec{F} \left(\vec{x}(t(\tau))\right)\cdot \frac{d\vec{x}}{d\tau}(t(\tau))d\tau$$ This seems to crucially rely on $\frac{dt}{d\tau}(\tau)$ being positive on $I'$ or equivalently $\frac{d\tau}{dt}(t)$ being positive on $I$, which I took to be the definition of $\tau(t)$ being an orientation-preserving reparameterization of the regular curve $\gamma$. Since $\frac{d\vec{x}}{d\tau} (t(\tau)) = \frac{d\vec{x}}{dt} (t(\tau)) \frac{dt}{d\tau} (\tau)$, I have noted that this is equivalent to the tangent vectors being parallel at any given point along the curve.
My confusion then comes from what is meant by orientation.
It seems that the orientation of $\gamma$ could equally be defined as being along the direction that the parameter decreases (antiparallel to the tangent vector) instead of the direction that the parameter increases (parallel to the tangent vector). Both are compatible with the notion of orientation-preserving reparameterizations. Is orientation independent of parameterization? For simple curves, does the notation$\int_{\gamma}$ only refer to integrating over the image $\{\gamma(t)\vert t\in I\}$ or the family of parameterizations of the image $\{\gamma(t)\vert t\in I\}$ with a particular orientation? If so, it is more correct to write $$\int_{\gamma}\vec{F}\cdot d\vec{x} = \pm \int_{I}\vec{F} \left(\vec{x}(t)\right)\cdot \frac{d\vec{x}}{dt}(t)dt$$ where $\pm$ is chosen such that the orientation of $\gamma$ is in the direction of increasing $\pm t$?
For example, suppose I wanted to integrate some vector field $\vec{F}:\vec{x}\in\mathbb{R}^3\mapsto \vec{F}(\vec{x})\in\mathbb{R}^3$ along the straight line segment connecting $(1,1,1)$ to the origin, starting from $(1,1,1)$ and ending at $(0,0,0)$. If we define $\gamma:t\in[0,1]\mapsto (t,t,t)\in \mathbb{R}^3$, would this line integral still be denoted $\int_{\gamma}\vec{F}(\vec{x}) \cdot d\vec{x}$ or would it be necessary to write $-\int_{\gamma}\vec{F}(\vec{x}) \cdot d\vec{x}$ or even $\int_{-\gamma}\vec{F}(\vec{x}) \cdot d\vec{x}$, where $-\gamma:t\in[0,1]\mapsto (1-t,1-t,1-t)\in \mathbb{R}^3$?