Names that point at a different plant
Snow chrysanthemum is a Coreopsis. Sweet neem is not neem. Black cumin is a buttercup relative. Wood apple is two different trees. Alligator pepper and grains of paradise are the same plant sold in two forms. In each case the name is doing the work of a description and getting it wrong.
Names the authority does not recognise
The USDA taxonomy lists 'wild marjoram' as the only English common name for Origanum vulgare — not oregano. It lists no English name at all for Elettaria cardamomum. Trade words like oregano and cardamom are commercial conventions rather than names any taxonomic authority assigns, and they are none the worse for it as long as you know which is which.
Names that are accusations
Safflower’s Latin name means 'the dyer’s', and two of its English names — false saffron, bastard saffron — record centuries of it being sold as something else. Ilex vomitoria’s species name libels a plant that does not do what it says, and has discouraged people from a perfectly ordinary drink since 1789.
Names covering several plants
Mallow covers a root, a flower and a hollyhock. Verbena covers a lemon shrub and a bitter weed. Mint covers two species people cannot tell apart dried. Kuding is sold from two unrelated genera. In these cases the name is genuinely ambiguous rather than wrong.
What actually resolves it
A binomial with its author. Where TeaHQ prints one it is because the name is settled; where it omits one it is because it is contested. Fifty-one records now carry a citation to the authority that settled it.