This thread contains many fine examples of HUGE associations with the word "Organic".
In the world of Science and particularly the discipline called Chemistry, "Organic" is the huge class of materials composed primarily of Carbon and Hydrogen atoms, but certainly including molecules also containing Oxygen and Nitrogen plus loads of other atoms (e.g., Sulphur, Phosphorus, Sodium, Potassium, Calcium, Iron, Magnesium, etc., etc.) More generally, these are all the molecules common in LIVING organisms, plant or animal. Their impacts on such organisms range from Important or even Essential to Harmful or dangerously toxic or destructive. Two VERY large factors in all of the impacts are the CONCENTRATION of the material, and the EXPOSURE TIME. Of course, those same two factors are vital in assessing the impact of NON-Organic materials on living organisms.
HOWEVER, in the modern world we have adopted a very different use of the word "Organic". This began with discoveries several centuries ago that certain materials derived from plants and animals had beneficial effects on people and other animals. Citrus fruit juice (we know now due to Vitamin C) to prevent Scurvy, Lanolin from sheep for softer skin - the list is vast. In the past century we accepted a major marketing trend to believe that ANYTHING that occurs NATURALLY in nature MUST be harmless, and probably beneficial. Sheer Folly! But the misuse has become almost an article of Faith. So anything extracted from a plant MUST be good, and anything MADE by people must be bad!
Enter Organic Chemistry. This is the sub-part of that field that specializes in these types of molecules. Simultaneously people in this field have found there is an ENORMOUS range of such molecules to be found everywhere in Nature, AND one can find ways to MAKE these molecules synthetically in a lab or factory without needing to find and isolate them from the complex mixture of materials found in Nature. But as has happened continuously throughout history, MISUSE of the knowledge (either out of ignorance or of malice) and resulting harm has been "translated" into a simplistic rule that "from nature" is harmless, and "Man-Made" is dangerous.
One fine counter-example I recall from my career is in large-scale processing of trees for lumber and paper products. For most such uses the first stage is to remove the outer and inner bark to leave just bare woody tree trunks. Now in a living tree, the active growth area is the inner bark where most of the organic chemical reactions of "living and growing" occur. A significant part of those actions are to produce chemicals that are toxic to insects and to microorganisms like fungus - these protect the growing tree from attack. But when the cut-down tree is processed, the first stage produces a "waste stream" of that bark full of such materials, often mixed with water used to soften it. If that is discarded simply by dumping it onto land or into a waterway, those NATURAL protective chemicals are toxic to the normal microorganisms that degrade tree wastes into soil nutrients for re-use in nature. So these natural tree chemicals must be pre-treated to destroy them before that sludge and water can be released into the environment to undergo further natural degradation.
In the field of artificial sweeteners, it happens that the ones discovered early were a among a large class of molecules called Aromatics, which are found as minor components of petroleum and coal that originate from ancient plants. For reasons I do not understand, somehow certain of these molecules stimulate the taste sensors on our tongues in a manner VERY similar to what sugars do, and we experience "sweetness" even though these chemicals have NO nutritional value in the digestive tract. Now, it ALSO happens that SOME other chemicals among those Aromatics are known to be quite harmful in several ways - some as toxins to certain body organs, some as carcinogens that cause cancer. Since we do not understand how those effects occur, either, we are cautious about any and all Aromatic chemicals. That does NOT mean that ALL Aromatics are harmful and must be avoided. However, labelling them as "Coal Tars" creates the belief that they all MUST be avoided (perhaps banned?). That simply is not true.
What IS true, though, is that ANY newly-discovered (or made) material needs careful study before widespread use. That was NOT being done a century ago and more. We have become better at it, but sometimes fail in a rush to exploit new discoveries, whether for direct benefit to us and other living organisms, or for simplistic short-term financial gain.
A small note on "shellac". The classic material under that name is extracted from dead insects and is the major component of a clear wood finishing liquid. But the label has been used also for several other clear liquids that may be applied to other materials to achieve a smooth shiny surface. So I am not convinced that the "shellac" used for candies is the same stuff used for desks and tables.