How is "gaiden" different from "omake"?
Omake=extra, gaiden=side story. In typical manga usage, omake is used to describe extra chapters/content that are something like an extra page tacked on the end of a chapter, or an extra chapter that's included in a volume release. "Gaiden" doesn't really have an established usage in the manga space beyond its dictionary meaning, the author just included it in the chap titles here.
This Chapter 10 is just a normal publication in the series, it's not volume extras or small unrelated extra pages at the end of an existing chapter, so it's not an omake. However, rather than being part of the main plot, it just covers a series of short unrelated side stories, so the author called it "gaiden".
Basically, they're similar things, but not really equal:
some things are omake+gaiden (X.5 chapters detailing side stories)
some omake aren't gaiden (character art extras, volume release announcements, etc)
some gaiden aren't omake (this)