session_regenerate_id（）如何在内部工作？

Question

I am trying to secure sessions for users on a website. On the login flow, I am regenerating session id, to ensure better security (reduce chances of guessing) in terms of session fixation. I was trying to test a script for the working of session_regenerate_id(). There are two instances of it. They are:
1) Delete old session by using session_regenerate_id(true)

<?php
session_start();
if (!isset($_SESSION['times'])) {
    $_SESSION['times'] = 0;
    $_SESSION['generate'] = true;
} else {
    if ($_SESSION['generate'] === true) {
        session_regenerate_id(true);
        $_SESSION['generate'] = false;
    }
    $_SESSION['times']++;
}
var_dump($_SESSION['times']);
var_dump(session_id());
?>

The following is the output I got for it:

/var/www/public/goto/index.php:31:int 0
/var/www/public/goto/index.php:32:string 'q96btv7i1ukpb3cq8tskcgdeb7' (length=26)

On another tab (same browser):

/var/www/public/goto/index.php:31:int 1
/var/www/public/goto/index.php:32:string 'gspq91o1jbcrcc27b4fkm4qgd5' (length=26)

On checking the folder after opening the same page in another tab where, session files are stored, I found the session file associated only with the string: gspq91o1jbcrcc27b4fkm4qgd5

2) Just regenerated new session id by using session_regenerate_id()

<?php
session_start();
if (!isset($_SESSION['times'])) {
    $_SESSION['times'] = 0;
    $_SESSION['generate'] = true;
} else {
    if ($_SESSION['generate'] === true) {
        session_regenerate_id();
        $_SESSION['generate'] = false;
    }
    $_SESSION['times']++;
}
var_dump($_SESSION['times']);
var_dump(session_id());
?>

Output for first tab:

/var/www/public/goto/index.php:31:int 0
/var/www/public/goto/index.php:32:string 'odu7sttll7pp43suqu22laa755' (length=26)

Output on second tab:

/var/www/public/goto/index.php:31:int 1
/var/www/public/goto/index.php:32:string '27po8j6bu318fd3e69ug4m53i3' (length=26)

In this case, I found session files for both the session ids: odu7sttll7pp43suqu22laa755 and 27po8j6bu318fd3e69ug4m53i3 Thus, what the function says, it does. But, my question comes here (sorry for the long intro), in the first case, when using session_regenerate_id(true) , the cookie on the first tab contained the old session id, whereas the one on the new tab, contained the new session id. After opening the second tab, when I refresh the first one, I magically get 3 as output and the session id of the second tab. What I don't understand here is when I refresh Tab 1 , the cookie it sends to the server contains the old session id, which shouldn't be no longer valid (and also doesn't exist as file on the server), how do I get 3 as output?
How is the same session persisted, even when Tab 1 has no idea about the regeneration and the new session cookie?
Shouldn't the request when I refresh the page on Tab 1 not get me the newly generated session (on Tab 2)?
That would be having two sessions on the same browser, which is not what actually happens. How does PHP handle this on session_regenerate_id(true)

Please note: The following is my output for session from phpinfo()

Answer 1

Thank you @Charlotte Dunois and @AnthonyB for helping me out.
It was a mistake on my part.
Chrome Dev Tools doesn't seem to update cookie information in real time. After checking All Site data on Settings, I found that my domain had only one cookie associated with it and that was the one for Tab 2.
So actually, it does send the new cookie when I refresh Tab 1. There could be an exception as @AnthonyB said (comment).

On Dev Tools, when I closed it for Tab 1 and re-opened it for Tab 1, I found the Cookies tab to have the updated value. I could no longer see the old cookie value.

Found out, Chrome doesn't do live update on the cookies even the cookie info has changed (like it does on page elements, styles).
I think that wouldn't be required too.
Cheers to everyone.