How do I associate the right Contact Card under Users & Groups to my own profile?
You don't need iCloud.com to do this. Open the Contacts application, select your card entry, click on the Card menu, choose "Make this My Card"
I ran across this issue when restoring a backup to a new phone. It was showing another family member's contact as My Card under Contacts. iCloud wasn't even signed in.
I had to go into Settings > Mail, Contacts, Calendars. Scroll down to Contacts and select the correct contact under My Info.
iPhone 6 with iOS 9.