For those of you who don't know, the outfit worn at karate is called a gi. As an underbelt (anyone under blackbelt) you will wear an all white gi with the proper color belt indicating your experience level. During the summer months (May-August) you may wear the informal summer gi. This means your usual white gi, but without the gi top and a plain black or white t-shirt instead.
When a karate-ka gets their black belt, a few options open up. If you are the top (or only) instructor in a class you may wear all black. If not, you may wear half-and-half. Most black-belts choose to wear black pants and a white top, but the reverse is completely acceptable.
If you are leading a class you have to wear the full gi. The summer gi is just for students.
It is wise to bring your gi top with you to class even if you aren't planning on wearing it because certain techniques such as grabs, throws, etc. are best done with a sturdy gi top to avoid stretched out our ripped t-shirts.
Master Motobu is wearing the gi pants in the style that is often worn in Okinawa, especially before karate adopted the Jiu-jutsu gi as their official uniform. They are essentially shorts that end just below the knee. Remember, Okinawa is a tropical island.
Just remember that your gi should be comfortable, loose fitting, and worn properly to facilitate your training.