"Water doesn't actually do anything"
You've drunk a glass of water before and felt nothing. Nothing lifted, nothing cleared. So whatever the studies say about laboratories, in your experience water is just water, and the coffee at least does something you can feel.
The comparison is unfair to the water, and it's worth seeing why. The coffee does something you can feel because caffeine acts on the brain within twenty minutes or so and produces a lift you notice. Water doesn't produce a lift. What it does is stop something, and stopping something is much harder to feel than starting something. If the headache at three doesn't arrive, you don't notice it not arriving. If the flat hour is shorter and shallower, there's no moment where you think "ah, water". The absence of a cost is invisible in a way that the presence of a lift isn't.
There is a study that tried to catch the effect directly. Caroline Edmonds and colleagues gave 47 adults either a drink of water or nothing, and separately told half of them that water would help their thinking and told the other half nothing. Twenty and forty minutes later, the people who'd drunk water rated themselves less thirsty and did better on a letter-cancellation task, a simple attention test, regardless of what they'd been told to expect. Memory span and reaction time didn't change. So the effect isn't placebo, and it's on attention rather than on intelligence, which fits the flat hour. Natalie Masento and colleagues reviewed the field in 2014 and concluded that certain aspects of thinking and mood do respond to water, while pointing out that the studies are inconsistent in how they measure things and often don't measure hydration properly. I'd take that as a fair summary. The effect is real, modest, and mostly about attention and mood, not about being cleverer.
The honest objection is that a modest effect on attention isn't going to change your life, and that's right. This book isn't promising a transformed afternoon. It's suggesting that the afternoon you have contains a headache and a dullness that are partly self-supplied, and that the supply can be stopped with a glass at eleven and a glass at one.
The practical implication is not to judge the water by how it feels going down. Judge it by the afternoon a fortnight later. Chapter 1's exercise gives you a baseline, and the flat hour is the thing to compare.