What is a generally accepted definition of "curve" in mathematics?

There is no unified definition. Curves in differential and algebraic geometry are defined very differently, via parametric and implicit equations, respectively. While the two representations can be related under some broad assumptions (via the implicit function theorem), both subjects push the envelope beyond such relatability. Mature fields are driven by technical reach, not intuition.

While connectedness and differentiability requirements are common in the differential context, they are not in the algebraic one. Studying connected components and algebraic singularities is a big part of the job. This is why hyperbola is one curve. Even within the classical differential geometry, different authors make different conventions about how differentiable a "curve" should be, from infinitely, to twice, to once continuously, each possibly piecewise (although continuity is usually assumed). There is also an intermediate area of analytic, holomorphic and pseudoholomorphic curves that combines methods from both approaches, and has definitional variations of its own.

Even just continuous curves, once deemed "pathological", like the Peano curve filling a square, or the nowhere differentiable Koch snowflake, now have a field of their own, a part of geometric measure theory. The study of such fractal curves has a very different flavor, employing distributions and measure theory, than the classical differential or algebraic geometry.