Correct. Looking down two headings on the wikipedia page, you have:
If the Lyapunov-candidate-function $ V$ is globally positive definite, radially unbounded and the time derivative of the Lyapunov-candidate-function is globally negative definite: $\dot {V}(x)<0 ~\forall x\in \mathbb {R} ^{n}\setminus \{0\}$,
then the equilibrium is proven to be globally asymptotically stable.
The Lyapunov-candidate function $V(x)$ is radially unbounded if $\|x\|\to \infty \implies V(x)\to \infty $. (This is also referred to as norm-coercivity.)
Roughly speaking, the maximal set $\mathcal{B}$ (w.r.t. inclusion) on which you can show $\dot V (x) < 0$ would be the basin of attraction for the equilibrium. Note that this isn't quite precise due to some topological concerns (certainly everything within $\mathcal{B}$ approaches the equilibrium, but there could be other points in the basin of attraction for which there is no such $V$ and $\mathcal{B}$).
You could pick the control law as follows
$$
u(x) = -x_1 - x_2 - \frac{x_1^2}{x_2},
$$
but that is not well defined when $x_2=0$.
Instead one could make use of the fact that your proposed control law makes the system dynamics linear, with
$$
\dot{x} = A\,x,
$$
$$
A = \begin{bmatrix}0 & 1 \\ -1 & -1\end{bmatrix}.
$$
A Lyapunov function for such system can be found of the form
$$
V(x) = x^\top P\,x,
$$
with $P$ positive definite which satisfies the Lyapunov equation
$$
A^\top P + P\,A = -Q,
$$
with $Q$ positive definite. If $A$ is stable any positive definite $Q$ should also yield a corresponding positive definite $P$.
For example when setting $Q$ equal to the identity matrix yields
$$
V(x) = \frac{1}{2} (3\,x_1^2+2\,x_1\,x_2+2\,x_2^2) = \frac{1}{2} (2\,x_1^2+(x_1+x_2)^2+x_2^2),
$$
$$
\dot{V}(x) = -x_1^2 - x_2^2.
$$
Best Answer
Strictly speaking, $V(x) = x^2_1 + x^2_2-1 $ is not a Lyapunov function because it is not positive definite. But one can take $W(x)=V(x)+1=x^2_1 + x^2_2$. It is positive definite, radially unbounded and its derivative $\dot W(x)=\dot V(x)+0=-(x^2_1 + x^2_2)$ is negative definite; thus, the origin is globally asymptotically stable.