As Charles says no can do.
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You have already touched on the problem. Data is flowing from one page to next.
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Or stated another way. In Word documents there is no such thing as a "page". Pages are only a 'logical' concept imposed by the physical printer device and paper.
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The type of "change history" you want to maintain is a legacy from pages typed on a typewriter then sent to a printshop to be photocopied and distributed.
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It cannot be done in Word. In theory, if you are to maintain a page level change history, if new data flowed onto the page, the change history would have to be changed. Right?!
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In Word you can only maintain change history at the Word section level.
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If this change history box is an in house requirement, you will have to go to the manager responsible and patiently explain why it no longer can be done that way.
You can maintain a change history by some sort of logical subdivision in the document like a "chapter" or subsection of a chapter. But each logical subdivision will start (and end) with a "word section break" with it's own header. If you make the Word section breaks "next page" there will be lots of white space in the printout. If you make the Word section breaks "continuous", then the section change history block may not appear on the first page of the section because there is content from the previous section hence the printed header will reflect that section.
Ultimately, the question is, what is the primary way this document will be used. Is it primarily an online document viewed in a computer. Or is the computer copy only used to create a printed copy that is distributed to all users.
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You may need to bring a "persuader" to help convince the manager that they will have to change the way they think about this document and change history

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If the requirement is imposed on you by and external entity, like government regulations, find a quiet corner and beat your head against the wall until it starts to feel better ...
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Been there done that (back in the '90's)!
Good luck.