Answer: Well yes, and no. First and third are definitely two viewpoints. But there are more...
"First Person" is written with an "I" narrator, as if the story happened to you. Just as you said.
"Third Person" limited, is written in third person, but limited to a single character's point of view. No other character's thoughts or feelings may enter the narrative. This is the pov chosen for most short stories.
"Limited Omniscient," is written in third person, and limited to a single viewpont in any one scene, but is considered omniscient, because it shifts from one character's pov to another's at scene changes. This is the viewpoint chosen for most novels.
True "Omniscient" pov is the godlike view of a story told by a narrator who knows all, including all the characters innermost thoughts. This is the familiar pov of fable and fairy tale.
Finally, there is the "Camera-eye" or what is sometimes called the "Exterior Dramatic" viewpoint, in which no single character's thoughts are revealed and every part of the story is told only with action. This is the most difficult pov in which to write, but it forces the writer to produce images. It's a good learning pov for writers who are poor at description, but can produce cold and "unfeeling" stories unless you are very good at description.
"Head hopping" is usually defined as switching from one character's mind to another's during a single scene, usually without meaning to. The following is an example of "head hopping" taken from an early story of my own, so as not to embarrass anyone else.
"He thought she was the most beautiful woman in the world and she thought so, too."
As you can see, we are experiencing both characters' thoughts. That's hopping back and forth from one head to another.