Mori,
The default heart rate zones set by Garmin are calculated using a slightly different methodology than the Uphill Athlete methodology.
Once you’ve determined ypur AeT and AnT as described in Uphill Athlete, you can manually adjust the heart rate zones on your Garmin watch.
Regarding difficulty staying below Z3, that was my experience as well when I started out with Uphill Athlete. It was very frustrating, I had to maintain what seemed like a ridiculously slow pace on level ground to stay below Z3. I did need to walk at times, and some runs it seemed like any effort beyond a moderate walk would spike my heart rate. I stuck with it, and in a few weeks I saw progress. Now, after a but over a year of following the Uphill Athlete approach, I can maintain a respectable running pace in Z1 for hours. It works, just hang in there!