What causes different friend list messages?

Solution 1:

I added a comment to your question, just as a joke, but after research it is in fact the right answer.

When someone from your friendlist is finishing arena with a score of 3 or less you get the " just finished an Arena run." message, but when it is 4 or more, it notifies you their score too, so you were right !

The sources I could find where forums as it appears to be like that since the notification was introduced, and the patchnote isn't precise :

Social – When your friends complete an Arena run or get a Legendary card in a pack, you’ll be notified so you can tell them, “Grats!”