Have 'natural response' defined clearly?
Warning: These are mostly opinions, skip this if you search only something exact.

Your linked lecture text has the following idea: Students should learn a few most common linear systems first as abstract concepts. The writer claims that mastering the math of those systems pays off because one can apply it for the rest of his life. 

The lecturer really starts from a differential equation, not from any physical system. With the same logic the children in the elementary school should at first learn the numbers as logical constructions - for example as sets of sets or as a declared, purely axiomatic system. In the beginning the only connection to the physical world: These objects (=numbers) can be used to present many relations in the world. The teacher says not a word about there's some connection between numbers and amounts or measures.

Your case: There's no law forcing the writers to use the same definition for term "natural response". Hopefully the writers stay consistent in their books and do not change the definition in the middle of their book. In homeworks and exams you should use the same definitions as your teacher.

I wouldn't use term response for anything which is not initiated by some external action nor doesn't characterize possible consequences of external actions. The evolution of the internal state of a system from the initial towards the final without any input has never been a response in my thoughts, speaks and writings. I would trash term "natural response" and say instead of it "autonomous state evolution". I suggest you to call a voltage over a capacitor a state variable in a system.

NOTE: I am not an academic person, only a practical tinkerer. That can cause some limitations to accept terms which are dissonant with everyday talkings I have used to hear.