Hi Undefined. Please see my above post for a summary of my ongoing uncertainty and query on this.
Hi Lakon. As usual lately, am rushed again today, but I'll try a couple of my own 'explanatory wrinkles' beyond those I mentioned already in my post to you, regarding the 'starting states' etc for the twin's motion thereafter (Remember I said that if the twins never started out from the same 'states' then they have no reference except their mutual Purely SR relative observations which cannot distinguish who is moving/faster and who is not etc....ie,
only relative closing/separation speed is observable in SR without any 'third party' information. ).
Anyhow, regarding the case where the twins START in the same motion states before any differential motion/acceleration is made by either one, then who is or is not accelerating/moving off will be an observable by making the rocket exhaust the 'telltale' indicator/information of which twin departs from the starting same-state 'standard clock/biological timing/aging rate conditions.
Another approach would involve creating an 'angle' between the path lines of one twin and the other as the 'telltale' information to distinguish who 'moved' and 'accelerated'. In this scenario, I would have the twins already in orbit around the sun, eg, along the same orbital path as the Earth).
Now have the 'intended' traveling twin leave (no rocket exhaust or other visible clue issuing from either rocketship), but leave with an acceleration profile such that his ship's orbit eventually changes to a SUN-POLAR orbit instead of the SUN-EQUATORIAL plane orbit! Now, let the traveling twin's ship finally enter/settle into an orbital 'diameter' and 'speed' which is the same as before (an Earth 'year' is still the same for both twins, except that the 'moved/accelerated' twin is now in a SUN-polar not SUN-equatorial orbit!).
The resultant situation will now have the twins passing each other twice every Earth/ship-orbit associated 'year' (which is still IDENTICAL for both twins), as their 'orbit circles' paths cross at 9-degrees because one 'circle' is aligned SUN pole-to-pole and the other 'circle' around SUN equator.
So the stay-put twin can definitely conclude that the other twin accelerated/moved because the traveled twin is passing him AT AN ANGLE, whereas before they were CO-PARALLEL (ie, moving alongside at same instantaneous speed in same instantaneous direction). Note that their 'speeds' are still the same, but their 'velocities' now differ because one has changed direction of orbit, and hence has been the one who accelerated/moved, according to the SUN-associated 'telltale' information/reference involving POLAR versus EQUATORIAL orbital paths and the 90-degree angles that results in.
But, again, without that polar/Equatorial EXTERNAL info, NEITHER twin can rely on the PURELY SR abstraction/model to enable them to distinguish who moved/accelerated; they would still only be able to observe a closing/separation speed and an angle between their now-differing orbital alignment.
In all cases of acceleration/moving from same-state conditions, only the moving twin can 'tell' he is the one accelerating. And only if he communicates his 'acceleration g-force experience' to the other stay-put twin will both have a common understanding of who is accelerating/moving and in what manner that now distinguishes between them absolutely.
So, to answer your specific concern/question, a purely SR abstract model/treatment STILL cannot distinguish between them UNLESS the EXTERNAL TO SR TREATMENT information (in this case, re the SUN poles/equator OR the INHERENT EXPERIENCE of g-forces 'information') is exchanged/understood between the twins.
Sorry. Gotta go, Lakon. I hope this rushed post isn't too typo-riddled to be clear on the essentials of what you wanted to have clarified at least! Don't know when I will be posting again. So bye for now, Lakon, and enjoy your discussions/explorations, everyone!