I think you want the top setting, which from what I can see on your profile is probably what you have selected now.
The word Public is in both settings and is a little confusing, but that basically means either way it's still on the internet inside imgflip.
The "Public and listed on imgflip.com" setting on the bottom means the stream name would be listed on your profile and the memes in that stream would be listed in a users profile page (and/or their Hot and Latest pages).