Start timer when button pressed (what is wrong with my code?)

I want a timer to start when my rocket takes off and stop when it lands. Stopping works fine (using a collider and state-switching) but I can't get the timer to start when I press the boost button.

Currently the timer text is displayed at the moment I pressed the button but clearly has been counting from the moment the scene started, not the moment the button is pressed.

Any help will be greatly appreciated. If there is a better way to start the timer rather than when I press the boost button then I am open to ideas!

Here is the all relevant code (I think). It appears as though it should be working to me, but it isn't.

// Timer
public Text timerText;
private float startTime;
private bool started = false;
private bool finished = false;

  void Start ()
{
    startTime = Time.time;  // get time since scene started
}

void Update ()
{
    if (currentState == State.Alive)
    {
        ProcessBoostInput();
        ProcessRotationInput();
        ProcessFiringInput();
        if (started)
        {
            RunTimer();
        }
    }

}

    private void ProcessBoostInput()
{
    if (Input.GetKey(KeyCode.A) || (CrossPlatformInputManager.GetButton("Fire1")))
    {
        BoostShip();
        PlayBoostSound();

        if (started != true)
        {
            started = true;
        }

    }

    private void RunTimer()
{
    if (finished)
    {

        return;
    }
    else if (started)
    {
        float timeSinceTimerStart = Time.time - startTime;

        string minutes = ((int)timeSinceTimerStart / 60).ToString();
        string seconds = (timeSinceTimerStart % 60).ToString("f3");

        timerText.text = minutes + ":" + seconds;
    }
}
private void StopTimer()
{
    finished = true;
    timerText.color = Color.green;
}