I think its belief in things that are obviously false - I'm a physics teacher, have a degree in physics, but am still a Christian, do I believe in creation as it is written in the bible though, nope. I can't rule out a greater power scientifically, but I can offer proof it didn't happen in the timeframes or way suggested by the good book.
Your DH believes the earth is flat, that is concerning because its easy proved to be incorrect, photographs from space, the fact that things vanish over the horizon as you move away from them, gravitational force meaning every planet is a sphere, not just us.
There are elements of the moon landing conspiracy that are compelling as presented in conspiracy shows, but there were so many people involved in the project that faking it, and keeping the secret decades later would be impossible.
The illuminati did most certainly exist, but not as a cult dedicated to world domination, more a boys club for artists, philosophers and scientists of its day - and there is no evidence of its continued existence, not a shred.
Most of the people presenting these theories are slightly edgy, mad individuals, by no means in the mainstream, that tells you all you need to know.
I'd be concerned about your husbands ability to fall down these rabbit holes and that he is easily conned.