First, a caveat. This is an industry with extremely fuzzy terms for roles and job titles. What is a UX designer in one group, might be considered the UI designer in another. What is considered the UI designer in one group, might be considered the lead UI developer in another. Etc. So, grain of salt and all that...
Secondly...umm...shoot...another caveat. ;)
Design as a broad term, and in an ideal process, permeates all team members. UX is design. UI is design. DB design is design. Coders are design. BAs are design. Etc. Again, idealistic, but it's something to strive for.
As such, I think it's hard to draw a solid line between 'Design' and 'UX'.
If we narrow 'design' a bit to perhaps 'graphic design' and 'ui design' or maybe even 'interaction design'. I think that makes things a bit easier, but still, there's no clear lines between these. There may be a LOT of overlap in some situations, and none in others.
I think it's probably easier to do the reverse. What tasks can you toss into which buckets? In your example, I think it's fairly easy to toss the 'user testing' into the 'ux' bucket.
Icon design? graphic design.
Form widget design? Interaction designer.
Again, all will still play a part, but there are likely defined tasks that go with each role.
So, in summary, I wouldn't say a UX professional has to have direct experience with user testing. But user testing, as a task, would likely have to go with the UX professional.