Does a price on the demand curve represent the highest or lowest price the consumer is willing to pay?
A distinction must be made between the individual demand curve and the aggregate demand curve.

An individual demand curve (or demand schedule) for a product/service, represents the pairs {price, quantity demanded} that co-optimize the consumer's utility maximization problem (together with all other quantities demanded of the other products that are present in the utility function).
It may appear that then this would be the "maximum" price, but it would be confusing to characterize it like that. Because, if the price changes, what would happen to individual quantity demanded depends on whether the good is normal or not, and in general whether there are "income-effects" or not (not all utility functions allow for income effects, by the way).
Moreover, at the individual level, the concept representing the consumer is quantity demanded, not price: the consumer observes prices and responds with quantity demanded.

Let's move now to the aggregate demand curve. It is simply the sum of individual demand curves for each price. Here we have two alternative approaches:  

a) The more "traditional" one is that all consumers have a non-zero demand schedule for all price levels, each lowering its demand as price increases. Here the previous discussion applies.

b) The alternative but equally "standard" approach is to argue that for each consumer there exists a (possibly different) "threshold" price, above which the individual demands zero quantity. So as price decreases, we imagine that more consumers enter the market with non-zero demand, and this is why as price decreases aggregate quantity demanded increases.
This approach implies that the price level in each price/quantity combination is the minimum ("at least") price for some consumers, while others would be willing to pay more (and so they have contributed positive quantity demanded for higher price levels).

I believe this is the approach taken by the textbook.