Historically, people have thought two places have been the start of the Amazon river. For the longest time, the Apurimac river was thought to be the start of the Amazon, feeding it from its own waters. Then, it dried up. Now, scientists believe it to be the Cordillera Rumi Cruz river. Well, that’s now the official start of the Amazon river, as the previous answer notes.
Throughout history, rivers have dried up. As rivers that feed into the beginning of the Amazon disappear or overflow, there will be a lot of doubt as to what the official start of the river is, despite science's best attempts to categorize it. With a river that has so many other rivers flowing into it, there will always be a river that’s slightly uphill from another, so the furthest uphill river is the head of the river.