I agree with your main point.
However, it’s not entirely true to state that “any carbon-based molecule is organic”. True, the study of chemistry is divided between “inorganic” and “organic” — but, there are other classifications.
The term “organic” suggests an origin in living organisms. There are a great many abiotic carbon molecules — i.e. not organic. /pedantry
One of the first lessons in Organic Chemistry is how to make an organic molecule in the laboratory from carbon and hydrogen.
Someone told me once that the definition of organic has changed because of legislation the redefines it. Kind of like how the government declared pi to be 3 and marriage to be a union between any two (or more?) people.