I've done some testing of my own, and I've successfully reproduced the problem.
First, I enabled PHP in Apache and created a dummy HTML page with a small PHP script:
<head><title>Sending mail with PHP</title></head>
Test of the PHP mail() function:<br><br>
echo ("Sending mail now, should be in your inbox in a few seconds.");
mail("firstname.lastname@example.org", "This is a test message", "This is indeed a test message", "From: Your friendly php script");
I accessed the page, and a few seconds later I had a new mail in my inbox (obviously, I specified a valid email address instead of "email@example.com").
I then edited php.ini and changed the sendmail_path setting to read sendmail_path = usr/sbin/sendmail
(note the missing slash at the beginning). I restarted Apache and refreshed the page. No mail was sent, and the following entry could be found in error_log:
sh: usr/sbin/sendmail: No such file or directory
Seems rather familiar. It looks like the setting you've been editing is either being overridden by another setting, or you're editing the wrong php.ini.
You could search for other files in /etc containing usr/sbin/sendmail
fgrep -r usr/sbin/sendmail /etc | less