Selecting a thumbnail image actually selects the whole section of the document containing that thumbnail. If you have not placed a section break (in a word processing document), the WHOLE document is in one section.
Not sure what action you take to "cut" the document. If you 'cut' it into two sections, then selecting and deleting a single thumbnail will delete all pages in the section containing the page for that thumbnail.
With word processing documents, Pages creates new pages as they are needed to accommodate the text you have entered in the body of the document, and the objects (such as images) you've placed within that text. To delete a single page, select all of the content on that page, then Delete. Note that this might be a two step process:
1. Select the text on that page, press delete. (other text from the page(s) below will flow up into the empty space.)
2. Select and delete the Objects on the page. (More text will flow up to fill the space, and the last page in the document will, if all text has flowed up from it, disappear.
In a Page Layout document, each page is a separate section, and selecting and deleting a single page thumbnail will also delete the section (page) that the thumbnail is an image of.
Regards,
Barry