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Microsoft 365 and Office | Word | For home | Windows
A family of Microsoft word processing software products for creating web, email, and print documents.
The official Watermark button is programmed to remove any existing watermark -- stupidly, from everywhere in the document -- before inserting the new watermark. The way to defeat this is to recognize that a "watermark" is really just a graphic (WordArt or a picture) that's inserted with its anchor in a header.
That means you can copy the existing watermark by opening the header pane of a page where it's displayed, clicking on it, and copying (Ctrl+C, or the Copy button on the ribbon, or the Copy command on the right-click menu). Then move the cursor to the header pane of the other page where you want it, and paste.
When you insert a cover page, you automatically get Different First Page layout (when the cursor is in the cover page's header, you'll see that checked on the Header & Footer Tools ribbon). That creates a separate header just for the first (cover) page. So if you have a watermark in the main document, you need to copy it from there to the first page header; or if the watermark is on the cover page, you need to copy it to the main header.