Discrete dynamical systems and continuous dynamical systems behave wildly differently. In particular, there are vastly more possible behaviors of discrete dynamical systems in low dimensions than there are for continuous dynamical systems. Prof. Strogatz is referring to the continuous case. The logistic map can produce oscillations of every period, and has periods of chaos. By contrast, chaos is not a phenomenon that's possible for continuous systems until you get to 3 dimensions.
Here is a proof that adding the condition $\nabla I(\mathbf x) \ne 0$ if $I(\mathbf x) = c$ makes the statement true. In fact, a stronger statement will be proved:
Let $M \subseteq \mathbb R^n$ be a nonempty, $k$-dimensional differentiable manifold, and suppose that $F(\mathbf x)$ is tangent to $M$ at every point $\mathbf x \in M$. Then, $M$ is invariant.
The lemma stated in the question follows from the latter due to the level set $I^{-1}(c)$ being an $(n - 1)$-dimensional differentiable manifold (this follows from the implicit function theorem when $I$ has nonzero gradient).
Now to the proof. Fix $\mathbf x_0 \in M$ and let $\varphi: U \longrightarrow V$, where $U$ is a neighborhood of $0$ in $\mathbb R^n$ and $V$ is a neighborhood of $\mathbf x_0$, be a diffeomorphism such that $\varphi(0) = \mathbf x_0$ and $\varphi(E \cap U) = M \cap V$, where $E = \mathrm{Span}\{e_1, \, \dots, \, e_k\}$. Any trajectory $t \mapsto \mathbf x(t)$ whose support lies entirely inside $V$ can be mapped to $U$ via $\varphi^{-1}$, i.e. we can define $\mathbf y(t) = \varphi^{-1}(\mathbf x(t))$. Then, $\mathbf y$ satisfies $$\dot{\mathbf y} = J_{\varphi^{-1}}(\mathbf x)\dot{\mathbf x} = (J_{\varphi}(\mathbf y))^{-1}F(\varphi(\mathbf y)) =: G(\mathbf y).$$ Now, whenever $\mathbf y \in E$ (which means $\varphi(\mathbf y) \in M$), $F(\varphi(\mathbf y))$ belongs to the tangent space to $M$ at $\mathbf x
= \varphi(\textbf y)$, $T_{\mathbf x}M = J_{\varphi}(\mathbf y)(E)$. Thus $F(\mathbf x) = J_{\varphi}(\mathbf y)\mathbf v$ for some $\mathbf v \in E$, and $G(\mathbf y) = (J_{\varphi}(\mathbf y))^{-1}J_{\varphi}(\mathbf y)\mathbf v = \mathbf v$. If we manage to show that the trajctories of $\dot{\mathbf y} = G(\mathbf y)$ starting at any $\tilde{\mathbf y} \in E$ lie (locally) in $E$ (that is, $E$ is invariant for $G$) we are done, since these trajectories are one-to-one with the trajectories of $\dot{\mathbf x} = F(\mathbf x)$ starting at $\tilde{\mathbf x} \in M$ via $\varphi$, and $\varphi$ maps points of $E$ to points of $M$.
To do this, consider any solution of $\dot{\mathbf y} = G(\mathbf y)$ with initial value $\mathbf y(0) = \tilde{\mathbf y}$. If we let $\tilde{\mathbf y} = (\tilde{\mathbf z}, \, 0)$ (where the $0$ is $(n - k)$-dimensional) and, for $\mathbf z \in \mathbb R^k$, $G((\mathbf z, \, 0)) = (H(\mathbf z), \, 0)$ (recall that $G(E \cap U) \subseteq E$), we can consider the system $\begin{cases} \dot{\mathbf z} = H(\mathbf z) \\ \mathbf z(0) = \tilde{\mathbf z} \end{cases}$ in $\mathbb R^k$. Due to the regularity of $H$, local existence and uniqueness of the solution hold. But now, if $\mathbf z(t)$ is the local solution, $\mathbf y(t) = (\mathbf z(t), \, 0)$ is a solution to $\dot{\mathbf y} = F(\mathbf y)$ that lies entirely in $E$, and it is in fact the only one.
Best Answer
A useful search phrase is “quadratic planar vector fields”.
Despite much research, there are still many open problems concerning such systems. For example, Hilbert's sixteenth problem asks for an upper bound for the number of limit cycles of systems of the form $\dot x = p(x,y)$, $\dot y = q(x,y)$, where $p$ and $q$ are polynomials of degree at most $n$. It is known that any given such planar polynomial system can have at most finitely many limit cycles; this was first claimed by Dulac, but it turned out much later that his proof was flawed, and a correct proof was found around 1990. But it's still unknown whether that finite number can be arbitrarily large or not (when you consider all possible systems of that form), even in the simplest nontrivial case $n=2$ (quadratic polynomials, as in your question). See, for example, the Scholarpedia article Limit cycles of planar polynomial vector fields.
There are of course special cases where you can say more, like the famous Lotka–Volterra predator–prey model where you have a constant of motion which causes all nontrivial trajectories to be periodic, but there's no hope of finding a general solution formula which covers all quadratic systems.