You can eat steak pink but not hamburger because the steak is dense and so it cannot get bacteria in it as easily as hamburger can.
Steak can be eaten rare because the bacteria cannot penetrate the steaks meat like it can with hamburger due to the steak meat being dense.
Bacteria can more easily penetrate the hamburger so that's why it's not good to eat hamburger rare and it should be fully cooked.
With a steak most of the bacteria is killed with the cooking process because the steak is dense which doesn't allow bacteria to penetrate the steak.
With burgers the bacteria can penetrate the burgers easily because it's not hard and dense like the steak is.
So that is why you cannot or are not supposed to eat hamburger rare when steak can be eaten rare.
A little pink in the hamburger is okay to eat but you should not eat the hamburger if it has a lot of pink because bacteria can be present in it.
Yes you can eat a burger with a little pink in it.
The burger with a little pink in it is safe to eat as long as you've cooked the burger to an internal temperature of 160 F throughout the burger.
If the burger has a lot of pink to it and not just a little bit then you need to cook the burger longer to remove most if not all of the pink.
Eating burgers with even a little pink is not very appetizing in my experience so I like my burgers to be cooked all the way through and not have a trace of pink.
But if it's just a tiny bit of pink to the burger then it's safe to eat but may not be very good to eat.