Garmin Connect website shows Obstacle Running under Running, but how can I enable this activity on the watch? The smartphone Garmin app doesn't show it either. ciq.forums.garmin.com/.../1367145.png
Not every 'activity type' that's available to tag an activity as in Garmin Connect is available as a native app on every watch. (Some aren't on any watch) If you do a Tough Mudder or something like that you have to record it using the regular Run app, or similar, and tag it as Obstacle Run manually in Connect.
you can add an activity - copy "running" and rename it "obstacle running" but not sure it will register as that when it downloads, but it will give you the ability to have certain parameters. for example i created a "treadmill walking" - it allows me to have the gps function automatically turned off.
Changing the name of an activity app won't change the activity type that it saves as. The main reason for doing that would be having different data screens set up for similar activities.
In the case of obstacle course I'd probably use the Trail Running app as a base that way the sections with obstacles where you're going slow with a higher heart rate won't affect your Vo2Max score.
No matter what you named your copy of Trail Running though it would still save tagged as trail running and you'd have to manually change the tag later in GC.