Let $W$ and $W^{\perp}$ be orthogonal Brownian motions. Next, define $$dS_{t} = S_{t}\sqrt{V_{t}}(\rho dW_{t}+\sqrt{1-\rho^{2}}dW_{t}^{\perp})$$
and define $X_{t}:=\log(S_{t})$. I computed via Itô's formula that $$dX_{t}= -\frac{V_{t}}{2}dt+\sqrt{V_{t}}(\rho dW_{t}+\sqrt{1-\rho^{2}}dW_{t}^{\perp})$$ which seemed rather simple, but then I came across the following paper: Affine forward variance models (Page 3, equation (2.4)), which—in passing—mentions that $dX_{t}$ is indeed
$$ dX_{t}= -\frac{V_{t}^{2}}{2}dt+\sqrt{V_{t}}(\rho dW_{t}+\sqrt{1-\rho^{2}}dW_{t}^{\perp})$$
I do not see where the $V_{t}^{2}$ comes from. Is it merely a typo, or am I missing something fundamental?
Yes, this is a typo. The dynamics you found for the log-price are correct and agree with the well-known log-price dynamics of classical stochastic volatility diffusions. The authors of the paper in fact use the correct formula when deriving the dynamics of another semi-martingale in Equation (2.22) on page 9.
For completeness, I include the derivation of the dynamics for $X_t$. Note thta ig $g(x) = \log x$, we have $g_x(x) = \frac{1}{x}$ and $g_{xx}(x) = -\frac{1}{x^2}$. For $X_t = g(S_t)$, Itô's formula gives: $$\begin{align*} dX_t &= \frac{1}{S_t}dS_t - \frac{1}{2S_t^2}(dS_t)^2 \\ &= \sqrt{V_t}(\rho dW_t + \sqrt{1 - \rho^2}dW_t^\perp) - \frac{V_t}{2}(\rho^2 dt + (1-\rho^2)dt) \\ &= - \frac{V_t}{2} dt + \sqrt{V_t} (\rho dW_t + \sqrt{1-\rho^2} dW_t^\perp) \end{align*}$$ which agrees with the formula you obtained.