This claim is true. You correctly identified the reasoning the article provides for why caffeine can both relieve headaches, and for why those who consume it more regularly and in greater amounts get headaches from caffeine because of withdrawal or overconsumption. However, your article lacks scientific references, or links to other pages with supporting evidence outside of its own website. I think finding a source that provides more concrete data, or from a trusted website/organization that specializes in medicine, could help back up your claim. I went searching for these sources, and in my source found a page from the National Library of Medicine, which more or less describes the same phenomenon that the original article describes--however, it goes into more depth and provides thorough data to back up its claims, as well as to fully explain the reasoning between the headache relief and pain caffeine has the potential to cause. One of the more notable aspects of the source is how caffeine promotes the loss of important nutrients from urinary excretion, which could also promote an increase in headache pain. Here is the article:
https://pmc.ncbi.nlm.nih.gov/articles/PMC10385675/#:~:text=on%20a%20group%20of%20200,factor%20by%202.4%25%20of%20patients.
In addition, here is an article from UCLA health that corroborates the statements about caffeine's affect on headache pain, as well as other potential health hazards that can result in headaches such as inadequate amounts of sleep, sleep apnea, high blood pressure, sinus congestion, and dehydration (caffeine can affect some of these factors as well). Here is the link: https://www.uclahealth.org/news/article/caffeine-connection-between-coffee-and-headaches
Overall, this claim is true; caffeine can cause headaches. However, more resources and more research is recommended to add to the original claim.
