I don't think that it has to be different each time. I count it as a win every time I send something out, even if it's not a brand new story.

I'm afraid I don't know what you mean by chain story though. Like a serial? I'd say contributing to one of those would count, since you're technically publishing each one by making it available to readers. Or perhaps writing short stories/vignettes set in the same locale or with the same characters? I do that pretty consistently with my Necropolis series, which is a set of short stories following an organization of Exorcists. Some stories are a few paragraphs long and others push into novella range.

Write one, Sub one is pretty forgiving, from what I've seen. The only defined part is that you have to write something, complete it, and submit it. And submission can be anything that puts a reader's eyeballs on your work.