Your distinction is correct, but I simply listed NSAIDs as an example, not an exhaustive list of pain relievers. You also make assumptions on “the normal headache pill” based on your locale. In the US paracetamol/acetaminophen/Tylenol is very common, but so is ibuprofen/Advil, naproxen sodium/Aleve, and aspirin/Bayer. In fact, I’d argue ibuprofen is far more popular here based on how much larger the ibuprofen section is compared to acetaminophen in pharmacies. Granted acetaminophen is a bit more common in compound OTC meds like cold and flu medicines.

That being said, paracetamol functions extremely similarly to NSAIDs, but it’s not anti-inflammatory, and works on the nervous system only, whereas NSAIDs affect the brain and body as well.