How to connect Big Dome Button with arduino
As others have said, the 12V powers the switch's internal LED.

EDIT:
Switches don't need power. If you skip all power to the switch it will still work but it won't light up.
End of edit:

If you want the switch to light up, connect the V in to 12V, or if you connect it to 5V, it will be dim, but -probably- still light. (You are certain to have a 5V pin on your Arduino board. Using that means you don't need a separate 5V supply.)
Now connect one lead of the switch to a digital input, and the other lead to ground.
Then set your switch pin to INPUT_PULLUP mode. Now your switch will read HIGH until it's pressed, then it will read LOW.
create a variable at the top of your code called buttonState:
byte buttonState = HIGH;

Also define an unsigned long variable nextReadMillis:
unsigned long nextReadMillis = 0;

and a pressCount variable:
unsigned int pressCount = 0;

And define a debounceTime:
#define debounceTime 50

In your loop, do something like this:
void loop() {
    //Get the new millis(value)
    unsigned long newMillis = millis();

    //only look at the new button value if enough time has elapsed.
    if (newMillis &gt;= nextReadMillis) {
       byte newButtonState = digitalRead(buttonPin);

       //Only do something if the button state has changed
       if (buttonState != newButtonState) {
          buttonState = newButtonState;
          nextReadMillis = newMillis + debounceTime;
          //If the button is now in the pressed (LOW) state, increment the count
          if (buttonState == LOW) {
              pressCount++;
          }
       }
    }
}

That's pseudo-code. It likely has syntax errors. It's not meant to be copy/pasted. Use it as a guide for your project.
The code using millis() is a version of &quot;software debouncing.&quot; Electrical switches tend to &quot;jitter&quot; or &quot;bounce&quot; between their on and off states very rapidly when they are first pressed/released. By only checking for state changes after a brief delay, you ignore the jitter in the readings.
Note that if you want your LED to only light when the button is pressed, you could control it with another digital pin. You'd assign a different pin to controlling the LED output, set that pin to OUTPUT mode, and switch the pin based on button state inside the body of the if (buttonState != newButtonState) if statement.
If your LED draws ≤20mA at 5V, you could connect it directly to your output pin. If it draws more than that, you will need a transistor to drive it so you don't draw too much current from your output pin and burn it out. Wiring a transistor driver is beyond the scope of this post however.