I am seeing these definitions in the $\mathbb{R}^n$ context, not neccessarily on general manifolds. My definition of a Lie derivative given by:
$$[v,w]:= \frac{d}{dt}((g_v^{-t})_*w)|_{t=0}$$
where $v,w$ are vector fields on an open $U \subset R^n$, $g_v^t$ is the "phase flow" of $v$ and where the lower star index denotes the "push-forward map".
Before differentiating I am not sure what the expression $(g_v^{-t})_*w$ represents. As I understand, if we let $x(t)$ be the curve that satisfies $dx/dt=v(x)$ with initial condition $x_0$, then $g_v^{t}(x_0)$ is a solution to that problem. Hence, I suppose that for any point $x_0$ we get a new curve. Is this correct, i.e. is it true that $g_v^t$ is a map from $U$ to the "space of curves"?
Now, I have seen an argument that states that $g_v^{-t}(x_0)$ is then a solution to the same problem as before but now $dx/dt=-v(x)$. This is all more or less ok. I understand also the "general idea" of a pushforward map as an induced map. However I can not grasp what $(g_v^{-t})_*w$ is. What is such an induced map and what this composition mean?
See Wikipedia for push-forward of vector fields via diffeomorphisms (or you can also talk about pullbacks, which is just push-forward by the inverse diffeomorphism). If you want to stay in $\Bbb{R}^n$ and avoid manifolds, then you can view a vector field $w$ on $U$ as a smooth map $w:U\to\Bbb{R}^n$. Now consider a diffeomorphism $\phi:U\to U'$ where $U,U'\subset\Bbb{R}^n$ are open. Then, we can define the mapping $\phi_*w:U'\to\Bbb{R}^n$ as: for each $y\in U'$, \begin{align} (\phi_*w)(y):= D\phi_{\phi^{-1}(y)}[w(\phi^{-1}(y))] \end{align} So, you start with a point $y\in U'$, the target space of $\phi$. This now gives a point $\phi^{-1}(y)\in U$ in the domain. Since $w$ is a vector field on the domain $U$, we can apply it to get a vector $w(\phi^{-1}(y))\in\Bbb{R}^n$. Finally, you use the derivative $D\phi_{\phi^{-1}(y)}:\Bbb{R}^n\to\Bbb{R}^n$ (this is a linear map) and apply it to finally get another vector in $\Bbb{R}^n$. You can think of this as taking the Jacobian matrix $J_{\phi}(\phi^{-1}(y))$, at the point $\phi^{-1}(y)$ and multiplying it by the column vector $w(\phi^{-1}(y))$.
Notice how we started with a vector field $w$ on the domain $U$ and ended up with a vector field $\phi_*w$ on the target space $U'$. This is why it's called a "push-forward".
In the more general manifold setting, with a diffeomorphism $\phi:M\to N$, we replace the Frechet derivative $D\phi_{\phi^{-1}(y)}$ by the tangent mapping between tangent spaces $T_{\phi^{-1}(y)}M\to T_yN$ (denoted $T\phi_{\phi^{-1}(y)}$, or also called the push-forward linear mapping $\phi_{*,\phi^{-1}(y)}$, or also called the differential $d\phi_{\phi^{-1}(y)}$).