Was gonna say, I didn't know the car would let you take off from a standstill in anything other than 1st gear? I've never tried it, but I know if I'm in manual mode approaching a roundabout in 2nd and end up stopping, the tranny selects 1st gear uncommanded. So I just assumed it would override you.
(Edit: just re-read and you didn't say anything about moving from a standstill. I thought the shuddering was occurring while slipping the clutch at very low speed; once you're in motion the clutch plates are locked together and not slipping relative to each other and producing shudder. Please correct me if I'm wrong.)
(Edit again: you did say you're taking off from a standstill in 2nd in the original post. And also that 1st -> 2nd gear changes are jerky in D but smooth in M. Hmm, that's really confusing; as rondeo says it's the same robot performing both actions. Maybe your manual shifts don't match the RPM that the auto is shifting at?)
Not sure if your second gear test rules anything in or out. The clutch friction and take-up point to get rolling under 2nd gear will be different to 1st gear, which might account for why one works and the other shudders. I thought it might narrow it down to one clutch if you have shudder in 1st and Reverse but not 2nd, but I think Reverse is on the same clutch as 2nd?
Just came across this article. Seems a straightforward and correct summary for reference, and supports your hypothesis that you've got a clutch issue (after first ruling out the fluid):
https://axleaddict.com/cars/Powershi...mission-Faults