What is incorrect about this depiction of the hydrolysis of an ester?
There are indeed two mechanisms for ester hydrolysis, one using base catalysis and one using acid catalysis. The mechanisms of both are provided below (as an aside, you mention that your textbook isn't great- you should consider getting a different one, even if your teacher gives assignments out of the other book, it'll give you a second opinion and something to cross check your book against).

Base catalysed ester hydrolysis

Source: Organic Chemistry (Wothers and Clayden)
Acid catalysed ester hydrolysis

Source: Organic Chemistry (Wothers and Clayden)

Is the product really an acid? It doesn't even have an H+.
Yes, it is an acid. The acid is actually initially formed however because of the basic conditions of the base catalysed mechanism it gets deprotonated to give the carboxylate. Upon workup the carboxylate salt (in your example the sodium salt of the carboxylate) would protonate to give the carboxylic acid (COOH).
It's stated that hydrolysis can happen when esters are treated with an acid as well as a base. But acids only donate H+ so where would the extra O come from to produce an alcohol as a product?
The product is a gain a carboxylic acid, and as you can see from the mechanism above, water is involved, which is where the additional oxygen comes from.  Its worth noting that with the acid catalysed mechanism the whole thing is reversible.
It's stated that a molecule of water is produced (in the base catalysed mechanism). Is this statement incorrect, or is the diagram incorrect?
It could be water or an alcohol depending on whether hydroxide or the alkoxide does the deprotonation. Water is most likely however, as shown above. In the acid catalysed mechanism it is the alcohol that is formed.
In general, how would we change the diagram/information below to make it correct?
See the mechanisms above, taken from Clayden and Wothers. In general the diagram in your textbook looks essentially 'fine', though I dont entirely like the way the structures are drawn (its a very american textbook way of doing it.)