Historically, "function" meant something like an element of the vector space $C(\mathbb{R})$ of continuous functions $\mathbb{R} \to \mathbb{R}$, "functional" meant something like a linear functional $C(\mathbb{R}) \to \mathbb{R}$ (that is, a thing which takes functions as input and returns numbers), and "operator" meant something like a linear transformation $C(\mathbb{R}) \to C(\mathbb{R})$ (that is, a thing which takes functions as input and returns functions).
Of course, from the modern point of view there's no real reason not to call operators functions, since we recognize sets of functions as after all just sets so we can talk about functions in and out of them. It's just convenient in functional analysis to invoke a certain context by using the word "operator."
You're more or less correct about what an operation is.