Why is my intuition incorrect?
I posit your intuition is incorrect because you learned from a biased source.
You probably learned about the idea of independence from talk of (in)dependent variables introductory calculus. However, dependence there is not spoken in a general sense, but is instead oriented towards a very specific application.
Specifically, actual problems are often most naturally expressed in terms of related variables, but introductory calculus tends to be presented in a very function-oriented manner. Thus, one is taught to re-express such problems in terms of functions, the typical method being to single out one or more variables (the 'independent variable(s)') to be used as function inputs, and interpreting the remaining variables as function outputs.
The general definition of independence in this setting is actually of the following form: a collection of variables are independent if and only if the only function $f$ satisfying $f(x_1, x_2, \ldots, x_n) = 0$ is the zero function.
You can also talk about more nuanced cases of independence, such as continuously independent ($f$ is restricted to continuous functions), differentiably independent ($f$ is restricted to differentiable functions), analytically independent ($f$ is restricted to analytic functions)... and, of course, the case at hand: linearly independent ($f$ is restricted to linear functions).
Incidentally, the fact that independence can be expressed in terms of comparisons to zero is a sort of weird quirk that is often applied to simplify definitions; the point may seem more intuitive when expressed in the following equivalent form:
A collection $\{ x_1, \ldots, x_n \}$ of vectors is linearly independent if and only if, whenever $f$ and $g$ are linear functions satisfying $f(x_1, \ldots, x_n) = g(x_1, \ldots, x_n)$, then $f = g$.
