As private use we understand any use of TeamViewer for purposes that are neither directly nor indirectly paid. It is not about whether the service itself is paid but whether the service is rendered within the context of the creation of an added value with some kind of financial compensation.
Examples for an indirect payment:
a software company sells a software product for which it offers free support via TeamViewer
teamwork among colleagues within a company
use in associations and non-profit organizations with non-volunteer (i.e. paid) employees
access to a company computer for work-related activities
In the above mentioned cases TeamViewer is used for creating an added value, the payment takes place indirectly through salaries or a combination of sales/support activities.
In any case, a use of TeamViewer with family and friends without any financial compensation is considered private use.