The prime way to define analyticity of a function $f:\ \Omega\to{\mathbb C}$ is the one you have given: $\Omega$ is a region (i.e., a connected open set) in ${\mathbb C}$, and for each $z_0\in{\Omega}$ the limit $\lim_{z\to z_0}{f(z)-f(z_0)\over z-z_0}=:f'(z_0)\ \in{\mathbb C}$ exists.
There are several theorems describing or guaranteeing analyticity of functions $f$ expressed in particular ways, among them the following:
When the vector-valued function $(x,y)\mapsto {\bf f}(x,y)=\bigl(u(x,y),v(x,y)\bigr)$ is continuously differentiable on $\Omega$, and $u_x=v_y$, $u_y=-v_x$ for all $(x,y)\in\Omega$, then the function $f:\quad \Omega\to{\mathbb C}\ ,\qquad z=x+iy\ \mapsto w:= u(x,y)+iv(x,y)$ is analytic on $\Omega$.
When $\gamma\subset{\mathbb C}$ is a smooth arc and $\phi:\ \gamma\to {\mathbb C}$ is an arbitrary continuous function then the function $f(z):=\int_\gamma{\phi(\zeta)\over z-\zeta}\ d\zeta$ is analytic on $\Omega:={\mathbb C}\setminus\gamma$.
When an arbitrary complex sequence $(a_n)_{n\geq0}$ is given, such that $\rho:={1\over\limsup_{n\to\infty}\root n\of{|a_n|}}>0$ then the function $f(z):=\sum_{k=0}^\infty a_k\ z^k$ is analytic for $|z|<\rho$.
If $f(z)$ is an "analytic expression" using symbols like $z$ (i.e., ${\rm id}_{\mathbb C}$), $+$,$-$, $*$, $/$, $\circ$, principal value of $\sqrt{\cdot }$ or $\log$, $\exp$, $\cos$, ${\rm artanh}$, etc., then $f(z)$ is an analytic function wherever bona fide defined.
Concerning your final questions:
The function $f(z):=z^2$ is a polynomial in $z$, so it is analytic in all of ${\mathbb C}$. You also can prove this directly: ${f(z)-f(z_0)\over z-z_0}={z^2-z_0^2\over z-z_0}=z+z_0\to 2z_0\qquad(z\to z_0)\ ,$ and as this holds for any fixed $z_0\in{\mathbb C}$ we deduce that the derivative $f':\ {\mathbb C}\to{\mathbb C}$ is given by $f'(z)=2z$.
When the function $f(z):=z/\bar z$ where analytic in the given annulus then by the principle 4. above the function $g(z):=z / f(z)=\bar z$ were analytic there also. But you know very well that this is not the case; or you may check the CR equations for $g$.