Concerning the first question: you description is incomplete, even in the homogeneous case.

There are homogeneous geodesic metrics that admit smooth families of dilatations but are not made of flat Banach metrics. In particular, some Carnot-Caratheodory spaces are.

For example, consider the Heisenberg group $H$, which can be thought of as $\mathbb R^3$ equipped with the following group law: $$ (x,y,z)\cdot(x',y',z') = (x+x',y+y',z+z'+x'y) . $$ Observe that for every $t\in\mathbb R$, the map $\phi_t:(x,y,z)\mapsto (e^tx,e^ty,e^{2t}z)$ is a group homomorphism, and these maps form a smooth 1-parameter group of diffeomorphisms (and hence a flow generated by a smooth vector field).

Consider a left-invariant two-dimensional distribution $V\subset TH$ spanned by left-invariant vector fields $X$ and $Y$ whose values at $(0,0,0)$ equal $\partial/\partial x$ and $\partial/\partial y$, respectively. Equip this distribution with a left-invariant Euclidean metric. The distribution is completely non-integrable, so we get a Carnot-Caratheodory metric on $H$. Observe that $\phi_t$ maps $X$ to $e^tX$ and $Y$ to $e^tY$, hence it is a $e^t$-dilatation of the Carnot-Caratheodory metric.

The Carnot-Caratheodory metric is very different from Banach metrics. For example, its Hausdorff dimension equals 4.

Concerning the first question: you description is incomplete, even in the homogeneous case.

There are homogeneous geodesic metrics that admit smooth families of dilatations but are not made of flat Banach metrics. In particular, some Carnot-Caratheodory metrics are.

For example, consider the Heisenberg group $H$, which can be thought of as $\mathbb R^3$ equipped with the following group law: $$ (x,y,z)\cdot(x',y',z') = (x+x',y+y',z+z'+x'y) . $$ Observe that for every $t\in\mathbb R$, the map $\phi_t:(x,y,z)\mapsto (e^tx,e^ty,e^{2t}z)$ is a group homomorphism, and these maps form a smooth 1-parameter group of diffeomorphisms (and hence a flow generated by a smooth vector field).

Consider a left-invariant two-dimensional distribution $V\subset TH$ spanned by left-invariant vector fields $X$ and $Y$ whose values at $(0,0,0)$ equal $\partial/\partial x$ and $\partial/\partial y$, respectively. Equip this distribution with a left-invariant Euclidean metric. The distribution is completely non-integrable, so we get a Carnot-Caratheodory metric on $H$. Observe that $\phi_t$ maps $X$ to $e^tX$ and $Y$ to $e^tY$, hence it is a $e^t$-dilatation of the Carnot-Caratheodory metric.

The Carnot-Caratheodory metric is very different from Banach metrics. For example, its Hausdorff dimension equals 4.

Concerning the first question.

No there are geodesic spaces that admit smooth families of dilatations but are not made of flat Banach metrics. In particular, some Carnot-Caratheodory spaces are.

For example, consider the Heisenberg Lie group $H$, which can be thought of as $\mathbb R^3$ equipped with the following group law: $$ (x,y,z)\cdot(x',y',z') = (x+x',y+y',z+z'+x'y) . $$ Observe that for every $t\in\mathbb R$, the map $\phi_t:(x,y,z)\mapsto (e^tx,e^ty,e^{2t}z)$ is a group homomorphism, and these maps form a smooth 1-parameter grooup of diffeomorphisms (and hence a flow generated by a smooth vector field).

Consider a left-invariant two-dimensional distribution $V\subset TH$ spanned by left-invariant vector fields $X$ and $Y$ whose values at $(0,0,0)$ equal $\partial/\partial x$ and $\partial/\partial y$. Equip this distribution with a left-invariant Euclidean metric. The distribution is completely non-integrable, so we get a Carnot-Caratheodory metric on $H$. Observe that $\phi_t$ maps $X$ to $e^tX$ and $Y$ to $e^tY$, hence it is a $e^t$-dilatation of the Carnot-Caratheodory metric.

But the Carnot-Caratheodory metric is very different from Banach metrics. For example, its Hausdorff dimension equals 4.

Concerning the first question: you description is incomplete, even in the homogeneous case.

There are homogeneous geodesic metrics that admit smooth families of dilatations but are not made of flat Banach metrics. In particular, some Carnot-Caratheodory spaces are.

For example, consider the Heisenberg group $H$, which can be thought of as $\mathbb R^3$ equipped with the following group law: $$ (x,y,z)\cdot(x',y',z') = (x+x',y+y',z+z'+x'y) . $$ Observe that for every $t\in\mathbb R$, the map $\phi_t:(x,y,z)\mapsto (e^tx,e^ty,e^{2t}z)$ is a group homomorphism, and these maps form a smooth 1-parameter group of diffeomorphisms (and hence a flow generated by a smooth vector field).

Consider a left-invariant two-dimensional distribution $V\subset TH$ spanned by left-invariant vector fields $X$ and $Y$ whose values at $(0,0,0)$ equal $\partial/\partial x$ and $\partial/\partial y$, respectively. Equip this distribution with a left-invariant Euclidean metric. The distribution is completely non-integrable, so we get a Carnot-Caratheodory metric on $H$. Observe that $\phi_t$ maps $X$ to $e^tX$ and $Y$ to $e^tY$, hence it is a $e^t$-dilatation of the Carnot-Caratheodory metric.

The Carnot-Caratheodory metric is very different from Banach metrics. For example, its Hausdorff dimension equals 4.