Abso, yes i guess we see it slightly differently. I agree with everything you've said and agree that is natural mother should be everything you have stated. It's just that IME the natural tag is applied to plenty of as yet untested mothers, based on their historic fondness of children.
The mothers in my social circle are mostly mothers to toddlers and younger. Some of them have been labelled "naturals", some of them not. The naturals loved babies before they had one, and their entire existence is seemingly based on their mother status. And that is beautiful.
The thing is the non-natural mothers are every bit as good mothers to their babies, and are providing every bit as good care, but it doesn't define them and they have other strings to their bows.
It's clear to me that some of the naturals might not be so good with older children, whereas some of the non-natural mothers who are just as doting with their babies and doing just as well, seem like they might be better suited to it in the long run.
Perhaps my experience with the usage of the tag is not typical, and perhaps it;s means different things when applied to women of different ages or levels of experience. But in the context i've seen it used in recently, it is unfair to the non-natural mothers who are doing a great job, and love mothering just as much, but don't qualify for the natural mother status.