Are you referring to the original creditor account, or a collection?
If it is a collection, the status of any collection is "Open" provided the debt collector still has active collection authority, and is "Closed" if the debt collector no longer has active collection authority.
Collection authority terminates on an unpaid debt when either the debt collector has their collection assignment terminated by the owner, of if the debt collector owns the debt, they sell or assign it to another.
Collection authority terminates on a paid debt when the debt is paid, and thus there is no longer an active debt.
Some debt collectors have been noted to report a "Closed" status for periods where their collection authority has been temporarily suspended, such as by way of a timely request for debt valiation from the consumer, but that is not common.
Yes, the status should be consistent with its reporting to each CRA.
You can dispute and force consistent reporting.
It would only make a substantive difference in the case of a collection with the debt remaining unpaid, and it is also known that the collection authority has been terminated or the debt collector owned and has sold the debt. In those two scenarios, CRA policy instructs the debt collector to also delete their reported collection, so it could form the basis for a request to the CRA that they delete the collection per their own credit reporting policy.