Different Permalink Settings For Post and Page

You can solve this yourself with the means provided by WordPress.

Under Settings > Permalinks you define the appearance of the post URLs. There you write a /post/ in front of the individual URL.

For Pages you set the URL at the permalink of the page. There is no automatism for this. So you can simply set /page/ in the permalink settings for each page.

If you don’t want to bother with the pages, you can also add the following script to the functions.php of your child theme, your own plugin or a code snippet plugin:

function my_page_rewrite() {
	add_rewrite_rule('^page/([^/]*)/?', 'index.php?pagename=$matches[1]', 'top');
}
add_action( 'init', 'my_page_rewrite' );

function my_pages_permalink( $permalink_structure, $post_id ) {
	if ( empty( $post_id ) ) return $permalink_structure ;
	return ( "/page".get_relative_permalink($permalink_structure) );
}
add_filter( 'page_link', 'my_pages_permalink', 10, 2 );

function get_relative_permalink( $url ) {
	return str_replace( home_url(), "", $url );
}

I have only roughly tested this.

Thank you @threadi for your help.