Update: solved

After another very careful reading of the docs and looking at the source code of the plugin, I added the line CGI::Session->name('MY_SESSID'); and voila. Turns out if you change the name of the cookie in your configuration block, you have to let CGI::Session know about it or it creates another cookie/session, wiping out the old one. As soon as I added the line, all was good. I'm curious where it stores the name, because I'm not finding it in the actual /tmp/sessionfile.

package Super;
use warnings;
use strict;
use base 'CGI::Application';
use CGI::Application::Plugin::Redirect;
use CGI::Application::Plugin::Session;

(my $http_host  = $ENV{'HTTP_HOST'}) =~ s/(www.)([a-zA-Z0-9\-\.]+)/$2/
+;

$self->session_config(
                      COOKIE_PARAMS => { -name     => 'MY_SESSID',
                                         -expires  => '+8h',
                                         -path     => '/',
                                         -domain   => ".".$http_host,
                                       },
                      SEND_COOKIE   => 1,
                     );
CGI::Session->name('MY_SESSID'); #Added this line
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