I go to build a planetary defense and see that "No defense slot is available." I do have slots available on that planet, and the manual section on "Planet Yields and Improvement Slots" says: "Available improvements include Factory, Lab, Farm, Mine, Market and Planetary Defense Base." Nothing about planetary defense being somehow a different category of slot. What's the deal here?

Planetary defenses are a different category of slot; you can research up to three of them.

The technology that unlocks the first planetary defense slot is "Defense Grid." The rest are unlocked by "Defense Network" and "Floating Fortresses".

Yep. All planets can host up to three planetary defense improvements
The "slots" themselfes will only be visible, once you have researched the first level of the planet defense improvement

Thanks.

If I could suggest... "No defense slot available" might not be the most helpful message on the build screen. Maybe "Tech not researched" or maybe not even showing up as an option. Or alternatively, distinguishing between the two kinds of slots in the manual.