Sheldon Brown died in 2008; C6000 came out in 2014, so you won't be getting His wisdom on that site in relation to these.
2014: https://productinfo.shimano.com/download/?path=pdfs/archive/2014-2015_Specifications_v020_en.pdf
gives 6010 (high efficiency) - 8R (roller brake) & 8V (v-brake)
also 6000 (standard efficiency) - 8R, 8V, 8C (coaster-brake)
2015 spec adds -8D (center lock disc) & 8CD (coaster brake with special five-bolt disc)
2016 spec adds 1, i.e. 6010 becomes 6011. And 6010 is deleted.
If one refers to any of the -11 or -01 diagrams, the changes are clear:
https://si.shimano.com/api/publish/storage/pdf/en/ev/SG-C6011-8R-8V/EV-SG-C6011-8R-8V-4040.pdf
https://si.shimano.com/api/publish/storage/pdf/en/ev/SG-C6001-8D/EV-SG-C6001-8D-4131B.pdf
Y3EH98040 vs Y36V98030 Carrier Unit (01)
and
Y3EG98030 vs Y36W98030 (11)
the other change is hub axle unit
Y3EK98020 vs Y37N98020 (01)
Y3EH98050 vs Y37L98030 (11)
i.e. the hub axle unit & carrier unit were both changed because they were deficient in some way
Carrier unit is similar to this
Axle unit is similar to this
http://rideyourbike.com/shimanoaxleunit.shtml
The same parts were also updated on Di2 and Alfine
https://si.shimano.com/api/publish/storage/pdf/en/ev/SG-C6061-8C/EV-SG-C6061-8C-4039.pdf
https://forum.cyclinguk.org/viewtopic.php?t=64432&start=630#p1267269
So basically you can use either hub, but C6001 should be an improved version.