Big Fish mid core games are mostly the free 2 play and mobile games that allowed you to dabble when you felt like it and could put away and resume whenever you felt like it. At one point it was the future of gaming, but it did not take off as expected.
"...put away and resume whenever you felt like it" sounds more like casual gameplay than anything else.
You're probably right about what Big Fish's definition of "mid-core games" is, since they've been trying to introduce more games for mobile. But it doesn't help when most other websites that give a definition are using a different definition for what they consider to be "mid-core games" and asking Google comes up with different answers. It's definitely confusing.
Maybe mid-core could be considered "middle of our core audience" or "average of our core audience" -- which would be different for different game companies depending on which audience they're focussing on making games for or reporting on games for. Gamasutra and others are obviously writing for a different audience than Big Fish is making games for.