Organic doesn't necessarily imply living.
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.
This is particular to stellar astrophysics. Astronomers studying asteroids maintain the conventional definition.
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' :)
Organic compound simply means that there are carbon-hydrogen bonds or carbon-carbon bonds in a compound.
The view that this requires "life" was a legitimate theory back in 1800s and was disproved in the late 1820s. (See the wiki link for more details)