Just to confirm, the close focus distance of any lens stays the same regardless of whether it is on a FX or DX body?
Yes, the working distance in front of the same 1:1 lens is the same on either body (it is the same lens), and the lens can still do 1:1 magnification on either sensor. The DX frame simply crops the overall frame view smaller than FX. But the object can be the same 1:1 size.
Ignoring the crop size in that 1:1 way is in contrast to the common knowledge that we have to stand back 1.5x farther with same lens on DX, or use a lens with focal length 1.5x longer on FX, to see the same view. But that specifically compares the cropped frame size view (seen in the smaller and larger frames), where 1:1 specifically describes the object size in either frame, regardless of how it might be cropped.
In film days, 1:1 means the image is the same size on the film as the object in real life. Take the film out, and a 1:1 photo of a US penny measured the same 3/4 inch diameter either way, on film or in real life. A penny at 1:1 will measure 3/4 inch on 35 mm film, and on 8x10 inch sheet film (that is the real life 1:1 size of a penny). At 1:1, you need an object 8x10 inches size to fill an 8x10 inch frame.
With digital, we cannot take the film out and look at it, but it still means the same thing.... if we photograph a ruler to see it, a 24x16mm DX frame will capture a 24x16mm area at 1:1... Or a 36x24mm FX frame will capture a 36x24 mm area at 1:1. And the object size in that (possibly cropped) frame will be 1:1 (real life size) either way.
If we enlarge and print both the DX and FX image, enlarged the same degree, the FX print will be 1.5x larger than the DX print... but the penny or ruler at 1:1 will be the same size in both prints (an enlarged size, but same size).
But if we enlarge the smaller DX image more, to print it same size as the FX print, then the DX print shows the object larger, which mimics a telephoto effect (cropped, then enlarged to same size). But it definitely is enlarged more then, which has adverse effect on image quality.
There are always some that imagine the sensor frame size does not matter, that it only matters how many megapixels we capture either way. They are of course hugely wrong.
Pixels do not create the image of course, the lens creates the image. The pixels simply attempt to sample the existing image sufficiently to reproduce it digitally (that image which has a size on the sensor), and it is that lens image in the sensor frame that is significant, which we try to reproduce with pixels.