The LAPD adopted the motto “To protect and to serve”, and it was subsequently widely copied across the U.S. Besides a general mandate to protect the public at large, police are responsible for the wellbeing of anyone in their custody.
They do have a responsibility to protect people in their custody.
That is a PR slogan. That's like trying to say Google can't be evil because their motto is "Don't be evil."
The Supreme Court has explicitly ruled that police do not have a duty to protect citizens, and that ruling has been reinforced consistently at lower courts throughout the country.
No, that's not what they meant. The ruling itself is extremely explicit. And furthermore, that's how other courts have applied the ruling in the decades since.
It's simply factually incorrect to claim that the police have a legal duty to protect citizens simply because of a PR campaign that they launched, when there are volumes of case law that establish otherwise.
AFAIK. it hasn't been copied in that form very widely, instead, it was popularized by TV (Dragnet specifically, IIRC), and so is culturally attached to police in general, despite the vast majority of departments with any motto having different mottos.