This question has been answered many times. have you searched the site?
One very important factor is missing in your description: save format. What I write below is valid only for .odt. If your document is .doc(x), no chance because M$ Word is based on different principles and has no notion of page styles.
A page style applies to a sequence of pages delimited by an “explicit” boundary. This boundary is a special page break created by Insert
>More Breaks
>Manual Break
where you can choose which page style to apply after the break. This page still is active until the next boundary.
This is for the basics. Page style switch can be automated in various ways.
In the case of Chapter Page → Default Page Style, configure Chapter Page so that it is always followed by Default Page Style. This is Next style parameter in Organizer
tab. Then when bottom of page is reached (by filing page with text) or when an “ordinary” Ctrl+Enter page break is met, switch to Default Page Style occurs.
Your chapter title is probably formatted with a specific paragraph style; e.g. built-in Heading 1. Customise Text Flow
tab so that a page break is automatically inserted before the paragraph, forcing a switch to Chapter Page. All you have to do is apply Heading 1 or Ctrl+1 to your title and everything is put in place.