Actually, if you have Outlook Social Connector enabled, it updates your contacts from GAL automatically. To check this, open View tab on the ribbon, click People Pane button and select Account Settings there.
The problem you're running into is that a Group is basically an alias for a collection of email address and contact objects; not a standalone object, and not just a list of strings containing email addresses.
So when you reference group, you're actually referencing all member objects in it.
Similar to when you send to a group, it doesn't send it to the group's mail box (cause there's no such thing), instead, the server receives it and sends a copy to each recipient in the group.
So you're saying "Move all messages received from people in this group to X", yet you don't want it to move when you receive email from people in the group. Your 2 wishes directly conflict.
If you want to do a plain-text search and move based on what the group is named, instead of what it represents, then use the text-in-header rule you've suggested, and live with possible false positives.
Unfortunately one gotcha is that, unlike a regular "from" rule, this rule will be client-side only. Meaning it'll only work when your copy of Outlook (that has the rule) is being actively used.
Additionally, in Outlook 2013 (I believe that's the first version with it) they've added a new rule that's very similar to the message Headers rule, only it's "with specific words in the sender's address".
Either will work, but the second will help avoid the false positives you suggest (i.e.: it being filtered because the address is in the subject, etc.).
Best Answer
I'm not sure I understand properly what you want to achieve, but here goes.. You have your personal Contacts folder in Outlook. That's the one that gets sync'ed with your smartphone.
Then you can have other other contacts folders (coming from Public Folders, SharePoint etc) that show up in Outlook only, and you can drag and drop from/to your personal Contacts, those contacts that you wish to share/add to your Outlook and sync to your phone. Right?
So you want to filter the contacts that get synchronized between your phone and Outlook? You can set filters (Right-click folder, Properties, Synchronization) on the Contacts folder to achieve some kind of filtering of contacts to your server, will that solve your problem? Otherwise, just leave all contacts in the Outlook Contacts folder synched with your phone, I have several hundred contacts on my phone, and I never have any sync problems. For the people in your organisation, you have the ability to search the Global Address List (GAL) from the phone, so you don't have to manage all those contacts manually.
Did I understand your request correctly?