Maybe this seems like a stupid question, but Wikipedia in multiple places will refer to hyperbolas, such as on https://en.wikipedia.org/wiki/Hyperbola , as a type of curve. Hyperbolas are not continuous over $\mathbb{R}^2$
On the page describing Moser's Worm Problem (https://en.wikipedia.org/wiki/Moser%27s_worm_problem), the term "curve" is used, and it seems like "curve" in that context is supposed to imply continuity.
Is "curve" just a poorly defined term that whose definition depends on the context? What am I missing?
Yes, what the word "curve" means depends on context. It turns out that the meanings in different contexts are very similar. Often, it is a "nice" $1$-dimensional subspace of a "nice" higher-dimensional geometric object.