From what I gather, you want to create a subdomain, e.g., webmail.example.com, via which your webmail software is accessible.
Further, you want webmail.example.com to remain in the browser's address field, even though the "actual" location of the webmail software may reside elsewhere, whether within Apache's document root or not.
There is an important proviso mentioned in the step of the tutorial to which you refer: "Also make sure that the vhost webmail.example.com does not exist in ISPConfig
[emphasis mine] (otherwise both vhosts will interfere with each other!)."
You don't need to create a redirect in ISPConfig, either.
Either you need to follow the tutorial exactly (don't create the subdomain in ISPConfig) or, if you'd rather manage the subdomain in ISPConfig, you need to:
1.) Create the domain
(not subdomain, as those are simply aliases) in ISPConfig, e.g., webmail.example.com. Again, do not specify a redirect of any kind for the domain.
2.) Create an A-record for webmail.example.com in your DNS configuration, wherever that might reside, and point it to your server's IP address.
3.) Create an alias for SquirrelMail in your Apache configuration.
Concerning step 3, add the following to the subdomain's Apache configuration in ISPConfig (Sites -> webmail.example.com -> Options -> Apache Directives box):
Alias / /usr/share/squirrelmail
It will take up to one minute for the Apache Directives change to take effect. You can review the change's status in ISPConfig (Monitor -> Show jobqueue).
Be aware that you may receive "Permission denied" errors when you visit webmail.example.com until you configure additional permissions.
But let us know when you get to that point...