What is 汉白玉 and is it 大理石?
From the links on your post:

但制作雕像的白色大理岩也称为大理岩
(But the white marble that's for making statue is also called marble)

'white marble' is one kind of 'marble'

'汉白玉'就是纯白色的大理石
'Han white jade' is pure white marble

汉白玉 is not actual jade, but a marble that is pure white

Is that translation correct? Is it a form of 大理石?
is there an English translation of this form of white marble?

Search 'white marble' image and you can find many pictures that we would call 白色的大理石, but most of them are not pure white.
汉白玉 should be translated as 'pure white marble' and it is still a kind of 大理石 (marble)
There is one pure white marble image I found online. It came from Hanoi
Hanoi is in North Vietnam, neighboring China
云石(cloud stone = marble) is another name for 大理石