The "whammy bar" goes by many other names as well: wham bar, wang bar, vibrato tailpiece, tremolo arm, Floyd, etc. (Although "Floyd" properly refers only to the "Locking Tremolo System" designed by Floyd Rose). The names come from manufacturers or from users.
Although some refer to this device as a "tremolo bar" or a "tremolo arm", the use of the word "tremolo" is misplaced. Tremolo refers to volume modulation. The term was originally used for instruments of the violin family. If a violinist's (or violist's or cellist's or bassist's) score is marked "tremolo", it means the player will rapidly move the bow back and forth across the string, resulting in a "trembling" sound.
"Vibrato", by contrast, is pitch modulation. Returning to the violin family: watch a violinist's left hand wiggle quickly while holding down a string. By moving her hand this way, the violinist is constantly and subtlely changing the pitch of the string. This makes the sound of the string very "wide" and "alive" -- more "vibrant."
The wham bar on the guitar changes the pitch of the gutar's strings. Therefore, it's properly referred to as a "vibrato" device, not a "tremolo" device.
You can use the whammy bar for anything from a subtle vibrato to a full on crazed dive bomb.
Is that finger vibrato, or is he using the whammy bar?