That link cannot help, since it is not public. It shows their own activities to anyone who looks at it. Make sure that the activity type is set to Jump Rope, and that the activity was recorded with a Garmin device.
As I wrote, those links are useless for the review, since they are not public. You have to open the concerned activity, make it public, and post a link to the detailed page of the activity, not to the list of activities!
No, those are just links to the activities list filtered by type. You have to post a link that ends with the numerical ID of the activity. Please see the document linked in my previous post explaining how you can share an activity.
When we follow those links you've posted we only see our own activities since there is no user identifier in the links. You have to post the direct link to the activity.
Since you got the first badge it should work if you log all the following activities in the exact same way.
I've got connections that got the Jumping Bean badge (10 activities) so it is working. I even got one that got the Committed Jumper badge.
I logged over 100 workouts logged with the Garmin Jump Rope app & activities, but also haven't seen even 1 respective badge appear. At least the new Jump Rope app can actually count jumps, unlike the hilariously inaccurate rep & step counters when using the cardio or strength jump rope activities. I guess it was too high a bar for us to expect Garmin to be good outside their runner wheelhouse. I guess I'll bin my Garmin jump rope expectations along side my Garmin soccer expectations & Garmin anaerobic expectations, hoping someday Garmin be good at tracking more than long distance running.
