Maybe posting your .htaccess can help figure out the redirect. There has to be a rule causing the redirect.

Well, I was able to fix the main issue by modifying the line that deals with the index.php. Here's my version:

<IfModule mod_rewrite.c>
    <IfModule mod_negotiation.c>
        Options -MultiViews
    </IfModule>
    Options +FollowSymlinks
    RewriteEngine On
   RewriteBase /my-project/
    # Redirect Trailing Slashes...
   RewriteRule ^(.*)/$ /$1 [L,R=301]


    # Handle Front Controller...
   RewriteCond %{REQUEST_FILENAME} !-d
   RewriteCond %{REQUEST_FILENAME} !-f
   RewriteRule ^(.*)$ index.php/$1 
</IfModule>

However, trailing slashes now cause a redirection to "not-my-domain.com/(URL)" if they are appended to any URL except the base URL.