1st LED turn ON when 1st button is pressed and other LEDs remaining OFF
First of all you have not implemented a stay on time of 5seconds delay(5000); but only 50ms delay(50);.

Second, you are not really switching of your LED's if no further button is pressed.

Change your loop() to the following, I guess this will help you to start your next steps into getting familiar with your code:

    void loop() {
      int stateButton_1 = digitalRead(pinButton_1); //read the state of the Button_1
      if(stateButton_1 == 1) {      //if Button_1 is pressed
         digitalWrite(LED_1, HIGH); //LED_1 on
         digitalWrite(LED_2, LOW);  //LED_2 off
         delay(5000);
      }
      digitalWrite(LED_1, LOW); //Switch off LED_1

      int stateButton_2 = digitalRead(pinButton_2); //read the state of the button 2
      if (stateButton_2 == 1) {
         digitalWrite(LED_1, LOW);
         digitalWrite(LED_2, HIGH);
         delay(5000);
      }
      digitalWrite(LED_2, LOW); //Switch off LED_2
    }


Hint: Using delay() is often not a good construct, better to do it with millis(), but this means some more changes. For the start in your case delay() will do, but check out what here in different Questions/answers is already written about delay() vs. millis()