This is precisely the question answered by the OP. The answer is, "because there is a whole spectrum of things you might mean by 'diversity', of which 'number of distinct species' is only one extremum".

And also, I assume, because the concept of "species" isn't all that well defined?

It is well defined: a group of living organisms consisting of similar individuals capable of exchanging genes or interbreeding

Ring species make your definition non-transitive. The same with species that can interbreed but exhibit hybrid breakdown.

I invite you to examine the notes of the international ornithological congress... The difference between species and subspecies is quite subtle, and subject to interpretation, because no one is really going to do the experiment to find out if two individuals of geographically district populations can actually still interbreed.

So if you have a few grams of soil and want to know how many species of micro organisms are in there, you're setting them up with dates to see which ones will end up breeding?

One of a number of definitions. It is one that allows lions and tigers to be the same species.

Does he provide an example of other definition of diversity that makes sense in biological context?

Yes, the two extremes are captured by the common metrics of "species richness" which is the pure "how many unique species are there", and "species evenness", which depends on how evenly distributed the species are. A community in which 99% of individuals are species A and the remaining 1% are from species B-G is exactly as species rich as a community in which there are equal numbers of individuals of each species, but it is much less even (and therefore, under one extreme of diversity, less diverse). In different contexts and for different ecological questions, these two different versions of diversity can matter more or less, and there are metrics which take both into account, but this is a fully generalized solution which shows you relative diversity along the entire spectrum from "all I care about is richness" to "all I care about is evenness".

-edit- by the way, since it may not be obvious to everyone, the reason why an ecologist might care bout evenness is because extremely rare species are often not very important to the wider community. From an ecological function perspective, there is very little difference between my above example of the 99%/1% community and a community that is 100% species A. So an community with two, equally populous species might have more functional diversity than a community with one very abundant species and several more, very rare species.