Ontology authoring is a complex process, where commonly the automated reasoner is invoked for verification of newly introduced changes, therewith amounting to a time-consuming test-last approach. Test-Driven Development (TDD) for ontology authoring is a recent test-first approach that aims to reduce authoring time and increase authoring efficiency. Current TDD testing falls short on coverage of OWL features and possible test outcomes, the rigorous foundation thereof, and evaluations to ascertain its effectiveness. We aim to address these issues in one instantiation of TDD for ontology authoring. We first propose a succinct, logic-based specification of TDD testing and present novel TDD algorithms so as to cover also any OWL 2 class expression for the TBox and for the principal ABox assertions, and prove their correctness. The algorithms use methods from the OWL API directly such that reclassification is not necessary for test execution, therewith reducing ontology authoring time. The algorithms were implemented in TDDonto2, a Protégé plugin. TDDonto2 was evaluated by users, which demonstrated that modellers make significantly fewer errors with TDDonto2 compared to the standard Protégé interface and complete their tasks better using less time. Thus, the results indicate that TDD is a promising approach in an ontology development methodology.