Intuition for the epsilon-delta definition of continuity

That is an almost correct intuitive formulation of what continuity is. Somehow you also need to get across that the actual size of the allowable deviations does not have anything to do with it. You could do that by saying "for any interpretation of the word 'small'", or something like that.

It does definitely show the general idea, though. Just for completeness, the general idea is formalised by the $\epsilon$-$\delta$ definition:

A function $f$ is continuous at a point $p$ in its domain if, for any $\epsilon > 0$, there is a $\delta > 0$ such that $$ |x-p| < \delta \implies |f(x) - f(p)| < \epsilon $$

The translation is that $\epsilon$ is the given bound on allowable variations in function value. $\delta$ is the bound you find on deviations from $p$ that keeps the function value within the given $\epsilon$-bound.