Great questions! The terms user testing and usability testing are often used interchangeably, which may be confusing.
User testing is an umbrella term that may refer to:
User research - discovering the potential user base for new products or services
Usability testing - testing a product developed with users to assess its ease of use
User acceptance testing - tests with users during which researchers assess the product developed against pre-defined criteria. UAT is carried out by the end of the product design process