Unfortunately this is not the case, although we have been led to believe so. Many autopsies state that a child has suffered encephalopathy from vaccination as the cause of death, and yet the infants death certificate is not changed to reflect that cause of death. It just remains as the blanket term SIDS.
A parent should not have to fight to have their childs death certificate reflect the evidence stated in their autopsy report.
I agree, however that SIDS should refer to exactly the instance you described.
This was how it was explained to my family in 07 as XBIL did not want an autopsy performed on his baby. He wanted her left alone. They were trying to express the need for the examination as an unknown cause of death in a baby is not readily accepted by child services unless they have excluded everything else for it to be classified as SIDS.