I believe "artificial selection" is the commonly accepted term.

Maybe he's a fan of The Far Side by Gary Larson?

https://www.amazon.com/Unnatural-Selections-Gary-Larson/dp/0...

And "breeding".

Controlled selection

I don't think humans are outside nature, it's a curious symbiotic relationship though.

That might be very true, but then the information content of the word 'natural' drops mostly to zero.

Since languages tend to drop words when that happens, we would need to find another word that conveys what natural means today -- at which point, why not just use the word as it is?

The word "natural" is a misnomer. For example you can buy "natural" fruit, despite the fact that a farmer planted and raised the plant, a chemist invented the chemicals used for pesticide, and a truck brought them to your farmers market. And of course all these things arose from natural systems being modified by us homo sapiens. Perhaps it's better to refer to things as person-made (or man-made if you must), to avoid the ambiguity of the word natural.

Given how deeply intertwined human society is with nature, any distinction between 'artificial' and 'natural' will always be somewhat arbitrary.

IMO, expecting precise definitions at this level is a lost cause.