Formal -- Imagine what you'd wear at someone else's wedding or evening ballroom parties, business meetings, etc. As you have written above, "formal" means evening gowns for ladies and suits and ties for the gents (sometimes referred to as "jacket-and-tie affair").
Casual -- as in "weat whatever the hell you want." This literally means ANYTHING, including jeans and T-shirts. You COULD wear suits and ties if you wish to, but the other way around (wear ANYTHING at "formal" events) is not true.
Informal, a.k.a. Business Casual -- is in the middle of the above two. Generally, this means anywhere between slacks to chinos with shirts WITH COLLAR (ties not required) for gents. This is the opposite of what you have written above (casual clothes WITH TIES), and though I have seen guys with jeans and ties, I don't think that's the norm.
The line blurs for the ladies, from dressy (but not necessarily classy) dresses or suits, to skirts / pants (as long as they're not inappropriate for casual business environment, i.e. denim, extremely short skirts, etc.) with shirts (not necessarily with collar).
Wow, it's easy... All I have to do is say "Agreed" to what Alexa has put so well. :)
Almost all the business-related parties that I've been invited to are "business casual" or "casual." That may be a regional thing - Seattle is pretty famous for being lax and casual. You'll even see people in jeans, if not T-shirts, at wedding ceremonies here. :)