Maybe I am missing some obvious setting in the AdminCP, but what do I have to enable to make these badges ("posts by...") visible in my forums?

It's a group setting "Highlight items with replies".

Thanks! That's exactly what I was looking for.

And it is always the badge that was uploaded within that specific group that is being displayed, correct?

Edited April 21, 20232 yr by MMXII

Yes, that is correct.