Disclaimer: I am a farmer who does certified organic production, according to the EU standard; I think other standards are mostly similar (if not outright interchangeable), but I can only claim to know about the EU one.
OK, right off the bat: the organic standard is not a management standard, but rather a production standard; what this means is that it won't give you any assurance regarding the methods used during the farming of the product you are buying/consuming. What it does is give you an assurance that the product complies with a set of rules, most of which are negative: don't use synthesized chemicals; don't use growth hormones; don't abuse fertilizers; don't use sludge; and so on.
This does lend itself to confusion. Imagine the subject of pesticides, herbicides, fungicides: yes, you can use products that serve this purpose; but 1) they must be approved to use in organic production 2) the amounts you can use are strictly regulated. It's true, you can use copper, or tea tree extract, or rotenone, or..; but for sure you won't (or you shouldn't) find traces of any of things like DDAC, glyphosates, etc. For many cultures, you need the help of those fungicides or pesticides to have a viable yield during your production cycle. In any case, every year I submit my products to a battery of tests to detect for more than 350 substances, and they always come with a clean bill of health.
The organic mode of production is a necessary condition for a sustainable agriculture, but it's by no means sufficient. At the very least, something like the implementation of a
GlobalGAP (which is a management standard for farms/agriculture) system is also a sine-qua-non condition for it. But, ultimately, knowing from whom you're buying from should be the litmus test of whether something was done properly or not.