False triggering on my Arduino Nano

This is my revised sketch, please check it out and see if I understood you correctly thank you so much.

// C++ code
//
int buttonState = 0;

void setup() {
  pinMode(5, OUTPUT);
  pinMode(10, INPUT);
  pinMode(7, OUTPUT);
  pinMode(1, OUTPUT);
  pinMode(11, INPUT);
  pinMode(2, OUTPUT);
  pinMode(12, INPUT);
  pinMode(6, OUTPUT);
  pinMode(3, OUTPUT);
  pinMode(13, INPUT);
  pinMode(4, OUTPUT);

  digitalWrite(5, HIGH);
  pinMode(5, OUTPUT);
  // BLOCKS LS. 3
}

void loop() {
  // START BUTTON
  buttonStatetate = digitalRead(10);
  if (buttonState == HIGH) {
    digitalWrite(7, HIGH);
    pinMode(7, OUTPUT);
    // BLOCKS START BUTTON
    digitalWrite(9, HIGH);
    pinMode(9, OUTPUT);
    // CLAMP CLOSING
  }
  // LS.1, CLAMP CLOSED
  buttonState = digitalRead(11);
  if (buttonState == HIGH) {
    digitalWrite(9, LOW);
    // CUTS POWER TO CLAMP
    delay(1000);  // Wait for 1000 millisecond(s)
    digitalWrite(2, HIGH);
    pinMode(2, OUTPUT);
    // POWERS MOTORS
    delay(6000);  // Wait for 6000 millisecond(s)
    // LS. 3 IS NO LONGER BLOCKED
    digitalWrite(5, LOW);
  }
  // LS. 2, MOTORS FORWARD
  buttonState = digitalRead(12);
  if (buttonState == HIGH) {
    delay(25);  // Wait for 25 millisecond(s)
    digitalWrite(6, HIGH);
    pinMode(6, OUTPUT);
    // BLOCKS LS. 2
    digitalWrite(2, LOW);
    // TURNS OFF MOTORS
    delay(2000);  // Wait for 2000 millisecond(s)
    digitalWrite(3, HIGH);
    pinMode(3, OUTPUT);
    // PUTS MOTORS IN REVERSE READINESS
    delay(1000);  // Wait for 1000 millisecond(s)
    digitalWrite(2, HIGH);
    pinMode(2, OUTPUT);
    // STARTS MOTORS IN REVERSE
    delay(3000);  // Wait for 3000 millisecond(s)
    digitalWrite(6, LOW);
    // LS.2 IS NO LONGER BLOCKED
  }
  // LS. 3
  buttonState = digitalRead(13);
  if (buttonState == HIGH) {
    delay(500);  // Wait for 500 millisecond(s)
    digitalWrite(5, HIGH);
    pinMode(5, OUTPUT);
    // BLOCKS LS. 3
    digitalWrite(2, LOW);
    // CUTS POWER TO MOTORS
    delay(1000);  // Wait for 1000 millisecond(s)
    digitalWrite(3, LOW);
    // PUTS MOTORS IN FORWARD READINESS
    delay(1000);  // Wait for 1000 millisecond(s)
    digitalWrite(4, HIGH);
    pinMode(4, OUTPUT);
    // STARTS CLAMP OPEN
    delay(15000);  // Wait for 15000 millisecond(s)
    digitalWrite(4, LOW);
    digitalWrite(7, LOW);
  }
}