Yes, /originally/ 'Organic' meant from living matter. However, when it was discovered that urea could be synthesised in the lab, we realised that there was no real distinction between 'living' molecules and 'dead' ones ...
Nowadays, 'organic' refers to chemicals with a majority (?) carbon content. It gets a little tricky for things like diamond and graphene (both materials) or for compounds that have a lot of heteroatoms.
Oddly enough, another nucleotide base (adenine) is exactly 50:50 C:N and has been proved to be synthesised from just HCN. So is it 'half'organic' :)
As a (former) organic chemist, this is spot-on. "Organic" doesn't really have a strict definition, but it's not so much "majority carbon" as it is presence of a C-H and/or C-C bond. I personally don't consider diamond and graphite organic, but some do. Carbon disulfide and phosgene (COCl2) usually aren't regarded as organic. And as you mentioned, stuff with carbon and lots of carbon atoms is a grey area (looking at you, azidoazide azide, but not looking at it too hard, it might not like that).
This is the difference between standard usage and jargon.
Chemists consider any covalent compound involving carbon that isn't only carbon to be "organic".
Normal usage is what you wrote.
Alternative but widespread usage is "agricultural products made without synthesized fertilizers and pesticides".
Oh, and business jargon uses organic to mean "natural market growth or internally developed capabilities", as opposed to buying the growth or capability.
And in military usage, organic means "a permanently assigned assisting unit", so that routine vehicle maintenance, for example, is handled by people who travel with the vehicle.
I've taken to using "organic-branded" for the agricultural products, since "organic" is essentially an FDA brand, with regulations for usage of the branding. Its also my personal rebellion against "organic" having a functional meaning of "magicalhealthygoodness" in some circles.
To their credit, I remember something about basically every element ionizes to some extent in space, thus exhibiting the "sea of electrons" behavior characteristic of metals.