It is impossible to give an exhaustive list$^1$ of Lagrangians for a relativistic point particle, but most of the physically relevant ones can be reduced from the following "master" Lagrangian$^2$
$$\begin{align}L~:=~&\frac{\dot{x}^2}{2e}-\frac{e (mc)^2}{2}, \cr \dot{x}^2~:=~&g_{\mu\nu}(x)~ \dot{x}^{\mu}\dot{x}^{\nu}~<~0, \cr \dot{x}^{\mu}~:=~& \frac{dx^{\mu}}{d\tau}, \end{align}\tag{A}$$$$\begin{align}L~:=~&\frac{\dot{x}^2}{2e}-\frac{e (mc)^2}{2}, \cr \dot{x}^2~:=~&g_{\mu\nu}(x)~ \dot{x}^{\mu}\dot{x}^{\nu}~<~0, \cr \dot{x}^{\mu}~:=~& \frac{dx^{\mu}}{d\lambda}, \end{align}\tag{A}$$
where $e=e(\tau)$$e=e(\lambda)$ is an einbein field, and $\tau$$\lambda$ is a world-line parameter (not necessarily proper time).
There is a corresponding Hamiltonian formulation as well, cf. e.g. this Phys.SE post.
The point is that this Lagrangian (A) is world-line reparametrization covariant, cf. e.g. this Phys.SE post. By fixing this gauge symmetry in various gauges, and perhaps integrating out some of the fields, the Lagrangian and Hamiltonian can take on many different appearances, cf. e.g. this Phys.SE post.
Examples:
If we integrating out the einbein $e$ in the Lagrangian (A), and go to thethe static gauge $x^0=\tau$$x^0=c\lambda$, we get OP's square root Lagrangian (1).
We could get a non-square root Lagrangian $\frac{m\dot{x}^2}{2}-\frac{ mc^2}{2}$ by going to the gauge $e=1/m$, cf. e.g. this Phys.SE post. And so forth.
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$^1$ This non-relativistic example should serve as a warning against trying to make a complete classification.
$^2$ We use the Minkowski sign convention $(−,+,+,+)$ and set the speed of light $c=1$.