I think there are two levels to the response I would provide. Under the Convention on Biological Diversity, there is a globally recognized definition for biodiversity that includes ecosystems, species, genetic diversity, and, importantly, sustainable use of biodiversity. That's at one level.
I very much agree with what I think is at the heart of your question and what really does drive people to be interested in biodiversity conservation. I think we need to think at scales that are more local or regional.
On species, people do identify with species in their backyard, if I can put it that way. In prairie Canada, farmers might relate to pronghorn antelope as a species they value. It's understanding at a regional scale how to manage habitat and sustainable development to deliver to people the outcomes they're looking for within their areas. If they value pronghorn antelope as well as the sustainable development around agriculture, how do you define that on a regional scale?
My short answer would be that I think the definition of biodiversity changes depending upon where you are, and it's really driven by personal values around the biodiversity that people have, if you will, grown up with.