What is this type of fixed point called? After numerically solving the following differential equations:
$$v'(t)=\frac{-\frac{1}{3} v(t)^3+v(t)-\omega (t)}{\tau } $$ $$  \omega '(t)= \frac{a+v(t)}{\tau}$$
at $a=1,\tau=0.2$ and taking $t$ from 0 to 10, I get this phase space plot:

The fixed point $(-1,-\frac23)$ is similar to an attracting focus, in that all nearby trajectories get attracted to it in a spiral. Note that the above plot is of a finite set of time steps, and there are no limit cycles here. 
However, as one gets closer and closer to the fixed point, it starts behaving more and more like a center in the sense that trajectories seem to form almost closed loops. Basically, the trajectories are reaching the fixed point, but asymptotically. 
As the nature of a fixed point is determined by the behavior of the trajectories in its immediate neighborhood, is this fixed point a center? Or is it  a stable focus? Or is it called something else entirely? (I've been calling it a "limit point")

Stability analysis can enlighten you here. Your system is determined by the a system of equation
\begin{align}
 \dot{\nu} = f(\nu,\omega)\\
 \dot{\omega} = g(\nu,\omega)
\end{align}
with a jacobian
\begin{align}
 J = \begin{bmatrix} \frac{1-\nu^2}{\tau} & -\frac{1}{\tau}\\ \frac{1}{\tau} & 0\end{bmatrix}
\end{align}
Near the fixed point $(\nu^*,\omega^*)=(-1,-2/3)$, it evaluates to 
\begin{align}
 J^* = \begin{bmatrix}0 & -\frac{1}{\tau}\\ \frac{1}{\tau} & 0\end{bmatrix}
\end{align}
with eigenvalues $\lambda=\pm i/\tau$. So, what can be said about this fixed point? It is a marginal case: eigenvalues have null real parts. In the case of interest the fixed point is called a center [Strogatz 2000, Nonlinear dynamics and chaos].
