Natural as in naturally occurring. Coal is formed and burns in nature with no interaction from humans. Plutonium does not occur naturally in detectable quantities. It only does so through human action.
Not sure what you’re trying to say? Asbestos is a natural mineral - not sure where you think that fits into this or how it’s relevant, sorry. We aren’t just listing all things that are naturally occurring!
I usually give the "natural" qualification if it is not immediately obvious there are plenty of naturally occuring sources of radioactivity around us. This has mostly to so with human perception of radioactivity being something unnatural, maybe it was not necessary on HN. It does not matter that much in the end with respect to radioactivity.