Gavgas yes you are correct but the damper and lower lube serves two different functions. The damper oil has anti foaming properties yes as the oil needs to pass through tiny holes on compression and rebound.
The lower lube is there for lubrication only.
The rock shock stuff used to be torco fully synthetic motorbike engine oil. I am not sure what they use now to be honest but it won’t be anything special.
A fully synthetic standard grade engine oil will be fine – you are right though, you do need to look out for any additive packs like EP (extreme pressure) as these can be aggressive towards nitrile / viton seals used. Just avoid any oils marked as anything special.