Everyone draws the line at a different place. For me, I prefer to avoid meat from livestock that were raised with growth stimulants such as added hormones or a newer class of drugs collectively called beta agonists (ractopamine also known as Paylean for pork and Optaflexx for cattle). Whether or not there's any risk from a food safety perspective or any impact on the livestock themselves, both can impact the flavor and texture of the meat, in my experience making beef blander and almost chalky. I also prefer to eat meat from livestock that have not been administered with what I'll call preventative antibiotics whether through their feed, a needle, or water. As for chicken, it's quite difficult in my area to find birds that have not been treated post mortem with brine/saline solutions but it's worth trying.

I'm not sure I fully understand this question. If you are specifically looking to avoid dioxins, then you'll want to limit the fats. Dioxins (if present at all) accumulate in the fat of meat & poultry animals as well as fish. This would apply to organic as well. Dioxins are in our environment. They are a product of our industrial world.

If that's not what you were asking, then I apologize.

I don't make purchasing decisions based on whether or not meat is organic, because I haven't seen anything to convince me that it will impact my health one way or the other.

I'm essentially living on a meat diet composed of my own product.

"Are certain meats safer to eat from non-organic suppliers?"

Well, obviously whole meats are far more safe than ground meats & ends. I really should buy myself a meat-grinder one of these days.