Understanding why race condition happens when only one thread does the write operation

I recently asked "Thrown object cannot be caught in a multi-threaded solution" and got the correct answer which works perfectly. However, I am still confused why there can be a race condition when only one thread does the write operation. Let me paste the original problematic code:

#include <iostream>
#include <thread>
using namespace std;

struct solution_using_thread {
    solution_using_thread()
     : alive_(true), thread_() {
        thread_ = thread([this]() {
            while(alive_);
        });
    }
    ~solution_using_thread() {
        alive_ = false;
        thread_.join();
    }
private:
    bool alive_;
    thread thread_;
};

int main() {
    cout << 0 << endl;
    try {
        solution_using_thread solution;
        throw 1;
    } catch (int i ) {
        cout << i << endl;
    }
    cout << 2 << endl;
}

Sometimes the output is only

0

According to the linked question, if I instead use member atomic<bool> alive_, the output becomes as expected

0
1
2

Now, I am trying to reason why member bool alive_ causes Undefined Behavior.

Case 1 (Happy ending):

• Variable solution is initialized:
  
  
  
  • solution_using_thread's default constructor sets alive_ to true in the main thread.
  
  
  • The thread starts and the value of alive_ happens to be true in the second thread. So thread execution is stuck in the while loop.
  
  
  • Before constructor returns, the second thread has already been started.
  
  
  
  


• We throw 1.
  
  
  
  • The destructor of solution is called. The value of alive_ is true in the main thread.
  
  
  • thread.join() blocks until the value of alive_ is synchronized with the second thread.
  
  
  • After some finite amount of delay alive_ is synchronized, the while loop terminates, the second thread finishes, the thread_.join() returns and stack unwinding is happily completed.
  
  
  
  


• The output is 0 1 2

Case 2 (Not desired, but at least not 'Undefined Behavior'):

• Variable solution is initialized:
  
  
  
  • solution_using_thread's default constructor sets alive_ to true in the main thread.
  
  
  • The thread starts and the value of alive_ happens to be false in the second thread. So thread execution ends immediately.
  
  
  • Before constructor returns, the second thread have already been started.
  
  
  
  


• We throw 1
  
  
  
  • thread.join() returns immediately because the thread has already finished.
  
  
  
  


• The output is 0 1 2

Obviously there is at least one more case in which it prints only 0. Can you describe that case?