You can customize your campaign to limit triggers to apply based on customers belonging to any particular list.  So in the case of the Birthday message, we'll send 1 message (with offer) to customers who have visited in the last 6 months, and a different message / offer to those who have not. 

To customize your campaign, navigate to Communications > Campaign Manager and click on Media Library.   Here you'll see a list of all the messages in your campaign.

Click on the Birthday message "Trigger" column so that the trigger details will appear at the bottom of the screen.    Let's start out by setting up the message for the frequent customers who will receive the Free Pizza. 

Use the "Limit To Customers From List" drop down to select the criteria of customers you want to reward generously.  Here we picked "Visited in the last 6 months".  

Save your trigger.    Check the "Offer" column to make sure the Free Pizza offer is associated with this message.

Now, we need to create a duplicate of this message for those customers who haven't ordered in the past 6 months - they will get the less generous "Free Appetizer" offer.

Click on the Birthday Message to edit it.  Change the Name to "BDAY - Happy Birthday (fewer visits)" (Customers won't see this name).  Click Save As New.

Once you save, this message should be selected in the "Add Additional Media" section at the bottom of the screen.  

Fill out the Trigger (Birthday) and Trigger Values as shown.   

We want to limit this message to go only to customers who have not ordered in the past 6 months, this will be the "free dessert" offer.    Use the "Exclude Customers From List" to choose your criteria for customers who will NOT get this offer.

Click "Link to Campaign". 

Now, from the campaign list, find your new message and click on the "Offers" link.  Select the offer you want to send to this segment of customers  - such as the Free Dessert.   

Now you have 2 different Birthday messages. Customers who've ordered in the past 6 months will get a Free Pizza, those who have not will get a Free Dessert. 

This technique is not limited to the Birthday message. You can use any sort of criteria to include or exclude customers from campaign messages. Considering using this to vary offers or rewards give to those who order online vs. those who don't, those who spend higher amounts per ticket (i.e. catering), etc.