It used to be the case that life was considered as some sort of fluid stuff which was added to the Elements to turn the 'inorganic' material into 'organic' material -- the low-level chemical distinction between 'nonliving' and 'living'. In Latin the life-fluid stuff was called vis vitalis and this view was called Vitalism.
What you are missing is that in 1828 a dude named Friedrich Wöhler took two indisputably 'inorganic' compounds, put them together, heated them up on a stove, let them cool, and got an indisputably 'organic' compound called urea, which disproved vitalism and established that 'organic' is a relatively arbitrary condition having nothing to do with life per se. Today it basically means "It contains carbon in it, and it wasn't one of the carbon-containing substances which were so plentiful in non-living places that vitalists considered them inorganic." The list of exceptions is not too long, but it dates back to this old philosophy. Organic no longer means 'living' because of the disproof of vitalism.
Chemist here. This is absolutely the correct response.
Organic does not imply biological in origin or nature whatsoever. The laws that govern organic chemistry are constant throughout the universe -- I'm not an astrochemist, but there is likely a nontrivial amount of very interesting organic chemistry happening in our solar system. I believe we've detected the presence of amino acids, for instance, which are the product of both organic (and in our case) biochemical reactions.
A lot of "organic" molecules are created spontaneously by inorganic reactions. Most of these molecules are important to living organism (and they (we) create more of these molecules on purpose, faster and more efficiently).
Slight correction and/or distinction here: organic molecules are formed by organic reactions. Organic does not imply biological, merely carbon chemistry. Addition, substitution, Diels-Alder, etc. etc. There is nothing special going on here, it's just how we organize and typify common themes and patterns.
Organic has a precise meaning to the chemist. See my other reply in the grandparent thread for further elaboration.
Surely finding organic material is an incredibly big deal, yet it's mentioned in passing. I feel as though I'm missing something fundamental here.