There is a well defined meaning of "insignificant" in this context: none of our activities have any material impact on what we see beyond the confines on Earth.

So the precise claim is that to describe any interplanetary scale (or up) process you can completely neglect humanity in its entirely, because its contribution to the process is negligible.

You are right that there's an unspoken step from "too small to impact the sun in any important way" to "insignificant to the universe", but it is not actually a very large step: being unable to affect one's surroundings is a classic sign of powerlessness.