Definitions of analytic, regular, holomorphic, differentiable, conformal: what implies what and do any imply that a function is a bijection?

Firstly, let me clear up the definitions for you. Let $U$ be an open set. Then:

An analytic function on $U$ (real or complex) is infinitely differentiable and equal to its Taylor series in a neighbourhood of every point in $U$.

A holomorphic function on $U$ (necessarily complex) is complex differentiable in a neighbourhood of every point in $U$.

A conformal function is a holomorphic function whose derivative is non-zero on $U$.

Avoid using the term "differentiable function" when working with complex functions. Use one of the terms above instead. I would also avoid "regular": it means the same as "analytic" but isn't well used.

For all functions (real or complex), analytic implies holomorphic. For complex functions, Cauchy proved that holomorphic implies analytic (which I still find astounding)! Hence conformal also implies holomorphic and analytic.

None of the definitions involve a function being a bijection, nor do any of them imply that a function is a bijection. You offer an example yourself: $f(z)=z^2$ from $\{z: \vert \mathrm{arg}(z) \vert < \frac{2\pi}{3} \}$ to $\mathbb{C}-\{0\}$ is conformal, hence also holomorphic and analytic, but is not a bijection.

However, in Latrace's answer to this question they prove that:

If $f$ is conformal in an open set $G$, then for each $a \in G$ there exists an $r > 0$ such that the restriction of $f$ to $D(a;r)$ is one-to-one (where $D(a;r)$ represents the open disk centered at $a$ of radius $r$).

... so in fact every conformal function can be restricted so that they become a bijection.