What you are trying to say is that a function whose graph can be traced without lifting the pencil is a common sense "interpretation" of continuity. But by itself it is not an interpretation of anything. It is only a damn good interpretation to you and I because we already understand continuity. To a student that is just starting the journey, it is nothing but an example of what we are about to delve into.
To put a point on all of this. Our common sense interpretation of common sense is wrong, as is often the case with common sense.