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What time is the child spending with each parent?
That makes sense. It is called a shared child arrangement order - where a child lives with each parent for a considerable part of the week/fortnight. If made as a child arrangement order it sets out the arrangement between the parents often with the help of a court order. So yes it does have a legal status.
The language which you refer to no longer exists. There are no longer residence orders or indeed shared residence orders. But there is no reason as to why from what you have written you could not agreed a shared child arrangement order.