But it does matter to their opinion of the site: is it down because the author took it down, is it down due to a technical problem, or is it is it down because the hosting provider took it down?

"This deployment is temporarily paused", if anything, sounds like the people who put the site up took it down again. That sends the wrong message.

Personally, if my hosting provider took my post down, I'd want them to make that obvious to my visitors. Or at the very least make it look like a technical issue. Not make it look like I took it down.