The reason I feel that there needs to be some sort of definition for a "genealogical statement" comes down to the analysis. I understand that information is primary or secondary depending upon the person's perceived knowledge of the event. It is the quantity of information contained in one document that is the problem. Not every piece of that information in that document is created equally.
A death certificate is the best example of the variation that can exist in one document. The informant on a death certificate may provide information on the parents of the deceased, the date of birth for the deceased, and the death date of the deceased. The analysis needs be done partially in the aggregate so that the researcher can tell how generally reliable the informant is--keeping in mind that an honest person can give incorrect answers if they've been lead to believe an incorrect fact their whole lives. That information provided also needs to be analyzed as separate pieces as that informant is providing primary information in some cases and secondary information in others.
Look at the information as a whole to see how generally reliable the person is. Look at the information piece by piece to see how likely the informant was to know that one specific piece of information.
It's a mistake to lump all that information the informant provided in one analytical pot. A child providing information on their parent's death certificate is not a primary informant for the names of the parents or the places of birth for the parents. The child would be a primary informant (at least probably) for when and where the deceased person died.
That's why I think it's good to ferret out each "genealogical statement" from the document, so that each statement can be analyzed on it's own perceived merits based upon the informant and their likely knowledge of the specific event.
But it's also good to look at the overall reliability of the informant--keeping in mind that they may simply be repeating information someone told them decades ago that they have no reason to doubt.