Civilization – Should I leave a space between a natural wonder and the settlement

It seems the manual is wrong and you simply cannot. It was probably possible in the original game but changed in one of the first few patches.

There is a mod that enables moving the capital.

EDIT: As of the wnter update, it seems it no longer works the same way. From the patchnotes:

Cities can no longer receive yields from more than one regional building per type; they take the highest (ex. production from multiple Factories)

Cities can no longer receive amenities from more than one regional building per type; they take the highest (ex. amenities from multiple Stadiums)

I came back home and tested it, it just so happens that I have a perfect example to answer this in my current game.

TL;DR : The bonus extends from the District itself, not the city center.

As you can see here, both Kwila and Kwango have industrial districts. By hovering over the city's production, I can see that Kwila receives a +6 from factory while Kwango only receives +3 from factory.

Both cities are 7 tiles from each other, and no other cities or districts of my own are within range of Kwila, so we must assume that Kwila is getting its +3 bonus from Kwango's industrial district, which is exactly 6 tiles away. Meanwhile, Kwango does NOT receive a +3 bonus from Kwila's factory, which is in the industrial district 7 tiles away (which is poorly placed, in hindsight), like the city center.

I will also assume this will work the same way with wonders, though I have no way to back it up right now. It also makes sense IMO, as it makes positioning of your districts and wonders a lot more important.