Thanks to Agnos for sharing all the information above.
I also had difficulties at first when trying to program the garage door opener. Turns out there were three problems, one my fault, two due to small errors in the VW documentation:
1) You have keep the two buttons on the roof of the car (buttons 1 and 3) depressed for 20 seconds before the Homelink device goes into the learning mode. 20 seconds might not seem like much when you read it, but it seems like it takes forever when you are holding the buttons. I was a bit impatient, and kept giving up before the lights started to flash. Moral of the story - look at your watch when you hold the buttons, after 20 seconds, the lights will flash.
2) As Agnos pointed out in his first post, the Phaeton does not have an indicator light above the Homelink control unit, as stated in the VW documentation - what happens after 20 seconds is that the illuminated numbers (1, 2, and 3) will all begin to flash slowly.
3) As Agnos also pointed out, you have to aim your existing handheld garage door opener at the front of the mirror base (see photo below), and not at the Homelink buttons on the roof of the Phaeton, as the VW manual suggests.
Anyway, once you get the numbers flashing slowly, release buttons 1 and 3, then press and continuously hold the button that you want to program, while frantically pressing and releasing your (existing) garage door opener - holding it in the correct position (see photo). When the Homelink unit finally gets the message from your existing opener, the three lights (1, 2 and 3) in the Homelink unit will begin to flash at twice the speed that they were flashing at when you first put the Homelink device into learning mode (end of step 1, above).
I also found out that it is handy to drive about a block away from your house before you begin the programming, otherwise, your garage door goes crazy from all the button-pushing.
I don't know if it is necessary to have the ignition of the car turned on in order to put the Homelink device into learning mode - Homelink suggests that the ignition should be turned on during this process.
There is an excellent illustrated, step by step guide available at the Homelink web site, at this link: http://www.homelink.com/program/gdo.html Note that if you purchased your pre-existing garage door opener transmitter (the 'clicker') in Canada, the last step of the programming technique is slightly different, because Canadian regulations limit the RF transmission from the handheld 'clicker' to a 2 second burst, regardless of whether you keep the button depressed longer than that or not.* Here is the link to the additional programming instructions for Canada: http://www.homelink.com/program/canada.tml
The Homelink website instructions suggest that the transmitter in the Phaeton can be put into learning mode by simply holding any single button depressed for 20 seconds or more. I tried doing this, and sure enough, after 20 seconds, the 1-2-3 lights all started to blink, suggesting that the device was ready to learn a new signal. I didn't bother completing the task, though, to see if it worked or not - I was grateful enough that I got it working according to VW's instructions (depressing buttons 1 and 3 for 20 seconds).
* If your Phaeton has been set up for Canadian operation by your dealer, then the Homelink transmitter is automatically limited to a 2 second transmission burst. This does not affect the programming (learning) process discussed here.
Where to hold the 'clicker' when programming the Homelink device: