Humans are complex beings, we use labels to describe as a way to think about that complexity. The real trick is to be able to use labels effectively without letting labels use you.

From what you quoted about wikipedia
a tendency to move between masculine and feminine gender-typed behaviour depending on context,
It is that whole "gender-typed behaviour" thing that messes with people. The truth is people have behaviours, preferences, beliefs, and ways of doing things. In order to simplify our interactions with people we as a society have classified certain things as being Masculine or Feminine. Strict adherence to these classifications causes all kinds of problems. My beloved wife knows way more about cars than I do, should she feel less feminine or I less masculine? I like wearing garters and stockings, does that mean I can't enjoy wielding a chainsaw? Obviously not.

I think that well rounded individuals ought to be able to be firm, and strong when need be, and soft and tender when need be, regardless of biological gender. A friend of my mothers is a retired admiral in the coast guard, he can operate a .50 caliber machine gun, has commanded men in battle during Vietnam, and does some of the most beautiful and intricate needlepoint I have ever seen.

So even if we find or create an answer to the question "What is bi-gendered" it does not mean that the term actually fits any one individual.

My preference when meeting people is to try to skip the easy labels and simply ask, "Who are you?"