This entry from wisegeek.com defines a boom truck as a truck that has an extending boom mounted on its bed. The device attached to the tip of the boom will determine what "kind" of boom truck it is - if it has a work platform, bucket, crane gear, concrete pump tubing etc, and what it can be used for will restrict the name that can be used to describe it.
I can give you the general definitions most used in my area.
A boom truck is any truck with a hydraulic boom that can be extended to perform a task. It can be used as a crane to lift objects, or as a bucket truck if the bucket attachment is installed on the end of the boom. Other accessories may also be attached to the end of the boom.
A bucket truck has a hydraulic boom with the bucket (usually permanently) attached for lifting a man in the air. Cherry Picker began as a brand name, but now refers to quite a few items used for lifting. If you enter the name into Google Images, you will see quite a few different examples, including bucket trucks, scissor lifts, and automobile engine hoists.