Chemistry - Difference between organic and inorganic chemistry
IUPAC is the International Union of Pure and Applied Chemistry, they make recommendations on the nomenclature. IUPAC mentions that the difference between organic and inorganic is not distinct. To quote "The boundaries between ‘organic’ and ‘inorganic’ compounds are blurred." in Brief Guide to the Nomenclature of Inorganic Chemistry R. M. Hartshorn, K.-H. Hellwich, A. Yerin.
Since the terminology of organic vs. inorganic is all human classification, it is not a binary system 0 or 1. What we can say now is that traditionally, all organic compounds do contain carbon. It can come from natural sources or purely synthetic. There is no such restriction. Plastic is an organic compound because it contains a lot of carbon chains.
Note that this word organic, as used in chemistry, has nothing to with the buzz word used in marketing of organic food, organic fruits, organically grown stuff. The word organic comes from French organique designating the jugular vein, hence related to organs or living beings.
It is a myth that Wohler started the distinction of organic vs. inorganic. There is a nice French article. The 16-paged paper is titled “Les origines de la chimie organique au-delà du mythe fondateur” (The origins of organic chemistry beyond the founding myth), Comptes Rendus Chimie, 2012, 15, 553–568.
Read the table at the end, and the abstract.
It clearly shows that Berzelius coined the term organic chemistry between 1807-1813.
Use Google Translate or DeepL to translate it.
Historically, organic chemistry was the chemistry done by living organism. It was (back in time) supposed to recquire some sort of divine element/intervention to be done, and therefore could not be done artificially (outside of a living organism). Since the discovery of artificial synthesis of urea in 1828 by Friedrich Wöhler, and the many more that followed, the definition of organic chemistry changed a lot.
Today, organic chemistry designate the chemistry of covalent bond. Those are mostly done by carbon and hydrogen atoms, and a few other hetero-atoms mostly oxygen, nitrogen and halogen. The way of reasoning in organic chemistry is to have a carbon skeleton, on which you will put functional group (which often will include hetero-atom). Despite not having carbon atom, you can consider the water molecule as an organic molecule (bonds are covalent). Some organic reaction can produce or consume water molecule, or need to be done in water. Same goes for $NH_3$.
Inorganic chemistry designate the chemistry of ionic and coordination bonds. It mostly focus on metal atoms. For a chemist, a metal atom is an atom that can lose up to 4 electrons (it is different from the physics definition, which defines metal by their macroscopic properties). A simple example of ionic bond is NaCl.
Coordination bond involve metal and organic molecule, therefore do mind that both fields are not exclusive. You can do inorganic chemistry with organic molecule (chelation of metal ions by ethylenediamine for example), or organic chemistry with metals (synthesis of crown ether using the template effect of a metal).
Plastics are organics polymers, salts can be seen as metal polymers (but nobody does). You can create polymers that use both organics molecules and metals, in which case you will do both organic and inorganic chemistry! (keywords for example would be "self-assembly coordination polymers", unfortunately all examples I can find come from publication behind paywall)
Finally, to go back to living organism, we know today that metal ions play a huge part in living organism. Bioinorganic chemistry is the field that study reactions of metal inside living organism. The early anti-cancer drug cis-palladium is the textbook example for this field.