Ha, I was confused too... but in the US, the terms coconut tree, palm tree and coconut palm are used interchangeably.
In India, you'd call a coconut tree a coconut tree and an arecanut tree a palm.
In India, you'd call a coconut tree a coconut tree and an arecanut tree a palm.
You're still confused. Coconut tree and palm tree aren't interchangeable; to be a coconut tree, the tree has to grow coconuts. All palm trees count as "palm trees", whether they grow coconuts or not. The prototypical palm tree grows dates.
This is a mix up on why certain oils are on the current “oils to avoid” list. Palm oil gets a bad reputation mainly because of environmental issues with how it's grown, not because it's unhealthy. But all these concerns get lumped together in the “bad oils” list.