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In this scenario, the Test Execution assignee would be the QA manager while the individual Test Runs would be assigned to one or more different persons.

Recommendations

• avoid cumulative testing assumptions because it may not be that trustworthy; sometimes you have a bunch of Tests that you executed once (some passed and some failed), then you create Test Executions just for the failing tests, assuming that the other tests still have the same result. As you may know, changes you make related with some faulty/incomplete requirement may affect other requirements; therefore, changes you make due some failing tests may implicitly affect other tests that supposely were already ok; 
• automate as much as possible, including your regression testing;
• take advantage of the fact that the Test Execution is an issue type; that means you can use workflows and take advantage of it in order to track the progress of the Test Execution. Xray provides some workflow possibilties for Test Executions as mentioned in Global Preferences

FAQ

Do I need to create a Test Execution every time I need to run the same Tests? Can't I update an existing one?

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You can also update the same Test Execution but you'll loose the benefit of tracking how your test results evolve through time.

How many Test Executions should I create? And when?

 Or... "How many Tests should you create?" There are no "correct" or unique answers for these kind of questions. You should create as many as needed in order to ensure the quality of the product you're working on. The quantity and the timing depends on the process and methodology you have implemented. As generic rules, you should start testing as soon as possible and avoid testing only at the end of your development cycle.