There are no descriptions that can be added to badges, no. These are indeed rules. For example the conversation one is as below. So you would add a rule to give the badge, and add the public description to that rule. In this manner, you can actually use a badge in multiple ways. Take for example you have a "helpful" badge. You may want to give that for 10 topics being marked solved, but also add it for when they hit 10 reputation points. 2 descriptions of why they got it, but only one badge.
The only other way a member can receive a badge is if you give them one manually from a post. In which case it will say "xyz for a post in a topic". You can add a reason for giving the badge when you do so, and that reason will appear on the post in question it is linked to