Is bicarbonate and bicarbonate of soda the same thing?

Yes, bicarbonate is used to refer to sodium bicarbonate, also known as baking soda.

If you're in chemistry class, bicarbonate has a more specific meaning. If someone told you to put a spoonful of bicarbonate into water and drink it to settle your stomach, they are referring to baking soda.

Baking powder is not the same thing. Baking soda is a base (the opposite of an acid). When mixed with an acid, it neutralizes the acid and creates a lot of carbon dioxide bubbles. If you remember the fake volcano that someone made in grade school, that's baking soda and an acid (usually vinegar), and the carbon dioxide bubbles cause the mixture to flow out of the volcano.

Baking powder is a mixture of baking soda and a dry acid. As long as they are dry, they don't reaction with each other. When they get wet, they react and create the carbon dioxide bubbles, and those bubbles are the difference between baking a biscuit and baking a hockey puck.

To settle your stomach, you want baking soda, not baking powder. Baking soda will neutralize your stomach acid. (Your stomach will make more acid over time.)