Right, selective breeding is just a subset of natural selection. Not sure why folks seem to want to exclude humans from nature, like we’re not part of the exact same system.
You’ve got it backwards. ‘Natural selection’ was coined to point out that selective forces exist in the wild other than humans deliberately breeding selected animals, which was well known in Darwin’s time. There’s ‘selection’ by humans and ‘natural selection’ which is by effects other than human selection. The effects of human hunting are a boundary case, where the selection pressure is not intended, but is a ‘natural’ consequence.
> Not sure why folks seem to want to exclude humans from nature
Probably because the concept of "nature" no longer serves any purpose if you don't, as it doesn't exclude anything.
It's often a pointless distinction to make (see also: anything that labels itself as "organic"), but it's a convenient shorthand to describe a category of interactions that people often want to talk about.
Perhaps, then, "nature" is a somewhat dated concept that is no longer especially useful to have in such broad circulation, given that it encourages folks to exclude humanity from analysis and understanding of the "natural" system?