There is a reverse protection diode between the DC socket and V-in, with a ~0.7volt drop.
6volt on the DC jack = ~5.3volt on V-in = ~4.3volt on the 5volt pin.
If you only have 6volt available, then it's better to connect that to the V-in pin.
Then the 5volt pin might just be 5volt.
Switch points are 0.3VCC and 0.6VCC.
If VCC is 5volt, then >3volt on an input pin should give a HIGH.
Tell us what you have connected to the input(s).
Leo..