I think the discussion can be solved pretty quickly... The core book mentions that there are Deathwatch Chaplains, but they're a special breed that need to be able to soothe the minds of all kinds of different chapters, with different beliefs so one could imagine them being more akin to an 'agnostic' or even buddhist, with an open mind for different views. They don't go into detail but I'd imagine a Deathwatch Chaplain either being recruited from the Deatchwatch members themselves, changing a temporary tenure into a permanent one and a conversion from 'trooper' to chaplain all happening 'in-house' within the Deathwatch.
Logic would be that a chapter-specific Chaplain will not quickly leave his normal post to join the Deathwatch as his duties and knowledge are of special value to the Chapter but I guess it would be not unheard of for the rare occassion that a Chaplain gets the urge or opportunity to join the Deathwatch. But in any case he'd have to temper his convictions in order to function as one unit with different chapter members.
So I think that the first option, the Deathwatch-specific Chaplain to be the more common variant, although very rare in terms of quantity.