The easiest way is probably to make a child theme and customize the results page as you want it, but it very highly depends on how the theme is actually coded. Some themes are built up from the back-end, so you need some sort of an API to access the output, others are just hard coded so you can change anything in the template files.
Because of that there is no generic approach as every single theme out there is different.
Let’s say you can do it via a child theme. In the theme directory the search.php file is the template called when outputting the search results. Making modifications to that is the solution in that case. You need to find the part where the actual text is printed and add a conditional statement to check which of the search instances has been triggered and redirected to that page, for example:
if ( isset($_GET['p_asid']) ) {
if ( $_GET['p_asid'] == 1 ) {
$text = "Text when results coming from search ID=1";
} else if ( $_GET['p_asid'] == 2 ) {
$text = "Text when results coming from search ID=2";
}
}If you want to avoid coding alltogether, then as an alternative solution you could create custom results page via the results shortcode. Then the results information box is going to be shown as the live search results are used as sort of a results page.