Cap
A French hat is commonly referred to as a beret. It is a soft, round, flat-crowned hat typically made of wool or felt and often associated with French culture.
I would call it a hard hat
The Bowler and the Derby is the same hat. The British call the hat a Bowler. The Americans call the hat a Derby.
A "top hat" is a type of hat that is often associated with magicians, so in this case, you would call it a "sitting top hat."
trilby
If it is soft just push it up.
A Panama hat.
beanie
The word is fedora. It is a soft felt hat with a wide curled brim.
It's called a homburg hat
rizal wears a bowler hat.
Tam-O-shanter / bonnet