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$^2$

$$\begin{align}L~:=~&\frac{\dot{x}^2}{2e}-\frac{e (mc)^2}{2}, \cr \dot{x}^2~:=~&g_{\mu\nu}(x)~ \dot{x}^{\mu}\dot{x}^{\nu}~<~0,  \cr \dot{x}^{\mu}~:=~& \frac{dx^{\mu}}{d\lambda}, \end{align}\tag{A}$$

where $e=e(\lambda)$ is an einbein field, and $\lambda$ is a world-line parameter (not necessarily proper time). 

There is a corresponding Hamiltonian formulation as well, cf. e.g. [this](https://physics.stackexchange.com/q/90552/2451) Phys.SE post. 

The point is that this Lagrangian (A) is world-line reparametrization covariant, cf. e.g. [this](https://physics.stackexchange.com/q/114862/2451) 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](https://physics.stackexchange.com/q/250911/2451) Phys.SE post. 

_Examples:_ 

1. If we integrating out the einbein $e$ in the Lagrangian (A), and go to the static gauge $x^0=c\lambda$, we get OP's square root Lagrangian (1).

2. 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](https://physics.stackexchange.com/q/130867/2451) Phys.SE post. And so forth.

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$^1$ [This](https://physics.stackexchange.com/q/17406/2451) non-relativistic example should serve as a warning against trying to make a complete classification.

$^2$ We use the Minkowski sign convention $(−,+,+,+)$.