Herewith a few possibly useful distinctions. Team v. Teamwork. Team is a noun and stands for some kind of mini- organization or entity that meets some kind of criteria, and that one can belong to. Best criteria that I know of are in Katzenbach and Smith, The Wisdom of Teams.
Teamwork is also a noun.
It stands for a pattern of behavior that can be observed, or found lacking, between two or more individuals, two or more work groups, two or more teams, various functions, departments, levels, etc. With or without supervisors or managers being involved. Teamwork may be highly desirable in most or all organizations and we OD types and trainers may spend a lot of time trying to develop it. But one can have lots of teamwork, but still not have teams, or be a member of a real team, which is an entity. Teams can be temporary, which most project teams and quality improvement teams and process improvement teams are, or permanent, although nothing is truly permanent.