One of the possible definitions for quasi-symmetry of a continuous injective map $\phi \colon \mathbb{S}^1 \to \mathbb{C}$ is the following: there exists $M > 0$ such that $$ 1/M \leq \frac{|H(x + t) - H(x)|}{|H(x - t) - H(x)|} \leq M, $$ where $H = \phi(e^{2 \pi i t})$, and $x \in \mathbb{R}$, $t > 0$ are arbitrary.
At the same time, we can consider $\mathbb{S}^1$ as the set $\{z \in \mathbb{C}: |z| = 1\}$, and use the general definition of quasi-symmetry between metric spaces: there exists an increasing homeomorphism $\eta\colon [0, +\infty) \to [0, +\infty)$ such that for all distinct points $x_1, x_2, x_3 \in \mathbb{S}^1$, we have $$ \frac{|\phi(x_1) - \phi(x_2)|}{|\phi(x_2) - \phi(x_3)|}{} \leq \eta\left(\frac{|x_1 - x_2|}{|x_2 - x_3|}\right). $$
Question 1: Do these definitions coincide (in the case of $\mathbb{S}^1$)?
It is known that if we have a quasi-symmetric map $\phi\colon \mathbb{S}^1 \to \mathbb{S}^1$, then it extends as a quasi-conformal and real-analytic homeomorphism of $\mathbb{D}$ (via Berlin-Ahlfors or Douady-Earl construction). Note that using the second definition of quasi-symmetry, we can consider quasi-symmetric maps between other Jordan curves. For instance,
Question 2: Let $C_1$ be a quasi-circle and $\phi\colon C_1 \to \mathbb{C}$ be quasi-symmetric. Is it true that then $C_2 := \phi(C_1)$ is a quasi-circle as well? And analogously to Berlin-Ahlfors and Douady-Earl the map $\phi \colon C_1 \to C_2$ extends to a quasi-conformal and real-analytic map between the corresponding open quasi-disks $D_1$ and $D_2$?
I believe that it is true and should follow from the following two facts: inverse and composition of quasi-symmetric maps are quasi-symmetric; conformal map between $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$ and quasi-disk extends quasi-symmetrically to $\mathbb{S}^1 = \partial \mathbb{D}$.
And similarly, it is known that quasi-conformal map $\varphi\colon \mathbb{D} \to D$, where $D$ is an open quasi-disk, extends continuously to a quasi-symmetric map $\phi\colon \mathbb{S}^1 \to \partial D$. That is why,
Question 3: If $D_1$ and $D_2$ are open quasi-disks and $\varphi \colon D_1 \to D_2$ is a quasi-conformal map. Does $\varphi$ extends continuously to a quasi-symmetric map $\phi \colon \partial D_1 \to \partial D_2$ (in the sense of the second definition above).
I believe that this also holds since it is true in the case when $D_1 = \mathbb{D}$ and due to the other arguments from the Question 2.