Learn English – the corresponding idiom “camel toe” for men

I am going to write this answer from a sociolinguistic perspective, because there is a lot at stake that can't be explained with a yes/no answer. Nonetheless we shall still make an attempt at giving a simple answer to your title question.

Yes "cop" is considered slang. No, it is not derogatory.

For a term to be considered derogatory, it has to indicate criticism or show disrespect. And under normal circumstances one would not use a derogatory term toward oneself because of its associated/inherent disrespect/criticism. But occurrences abound of cops themselves using the term "cop":

Florida cop yelling: "I'm a cop!"

Obama administration Department of Justice official in a conference with police chiefs from all over the U.S.: "If you throw a cop into a neighborhood and tell them to engage in enforcement activity..."

On top of the linked examples in your question that evince the prevalence of "cop" in journalistic writing, it is also widely heard in public talks, news broadcasts, and even police news releases. These are some examples I found on YouTube. Some of these took place in settings as formal as it gets.

A psychiatrist recounting what her police client told her: "I'm a cop. This is my job."

TED talk speaker telling a story.

David Cameron telling a story in a press conference with Barack Obama.

To sum, "cop" is not a slur, nor is it considered pejorative by most anglophones in this day and age. It could have been a pejorative term back in the day — way back, as per personal attestations of @BowlOfRed and @RBarryYoung previous to the 60's and 70's, — but not today. "Pig" and "pork" are pejorative terms for the police, and "the fuzz" and many others used to be too, not "cop".

Etymology

etymonline.com says:

"policeman," 1859, abbreviation (said to be originally thieves' slang) of earlier copper (n.2), which is attested from 1846, agent noun from cop (v.) "to capture or arrest as a prisoner." Cop-shop "police station" is attested from 1941. The children's game of cops and robbers is attested from 1900.

Wikipedia says

The term copper was the original, unshortened word, originally used in Britain to mean "someone who captures". In British English, the term cop is recorded (Shorter Oxford Dictionary) in the sense of 'to capture' from 1704, derived from the Latin capere via the Old French caper. The OED suggests that "copper" is from "cop" in this sense, but adds that the derivation is uncertain. Many imaginative but incorrect stories have come up over the years, including that cop refers to the police uniform's copper buttons, the police man's copper badge, or that it is an abbreviation for "constable on patrol", "constabulary of police", or "chief of police".

So "Addressing police as cops is disrespectful" is a myth? Why do some people keep saying that?

Insisting that cop is a pejorative term and perpetuating that myth help reinforce the public image of police being heroic and separate from the great unwashed. Despite what we are taught to believe: "No one is above the law", as a matter of fact the police are above the law and they are different from us legally speaking. Police effectively enjoy criminally extensive immunity that has allowed them to continue murdering people of color on the street without real consequences. I could point to numerous studies (most but definitely not all of them by African American thinkers) on this topic, but since it's way past midnight for me, I will just quote the New York Times to save time:

Police officers enjoy a web of protections against the consequences of their behavior on the job. From the legal doctrine of qualified immunity to state and local police indemnification laws, it is nearly impossible for a plaintiff to get any justice, even when an officer unquestionably violated his or her rights. (source)

^{Note: NYT is very generous with its wording. Black scholars have said a lot more damning things about this.}

The social message sent out to people via popular culture and news is that police are different from other people, socially, legally, and linguistically. Here is an interview where the creator of popular cop shows such as the Wire talks about how popular culture has helped put police on a pedestal. So to perpetuate this separation of police from the masses, the best sociolinguistic tool is to tell people to treat police with respect and call them "officers". It is interesting to note how the word "officer" comes from an earlier use in British and American military to distinguish commissioned military officials from enlisted men. The word "officer" itself has elitist undertones.

As for that Quora answer purportedly from a former dispatcher that claims police dispatchers never use "cop", I can tell you from experience that is not true. I have heard U.S. 911 dispatchers use "cop" several times. Different police departments carry out their training differently, so I wouldn't be surprised if in some agencies/districts dispatchers are trained to avoid this word, but it does not represent all American 911 dispatchers.

What is chillingly telling about that answer is how it shows you the us versus them mentality common among people in law enforcement. "Them" is the great unwashed, the masses, the ordinary folks, the policed. So to quote from the Quora answer you linked in your question (emphasis my own):

My instructor looked at me, "We don't use the word 'cop' to describe an officer of the law. We say police officer. They can use that word out there. We don't."

I got a lesson that day in dispatcher etiquette, my first taste of the us versus them mentality, and the feeling that perhaps "cop" wasn't the preferred term to describe an officer of the law.

Addressing police as "police officers" and insisting that "cop" is a bad word to be avoided is a very simple sociolinguistic tool to keep that social and linguistic distance.

Why did that person say such a thing to you? Did you get Karened?

I wasn't there to have witnessed it, so I can't be certain. But it sounds to me you had a racist and/or xenophobic encounter. Since we have chatted about similar things in the ELL chatroom, I understand the larger background, which is missing in your question. That person must have said those words on their own preconceived and stereotype-based notions about you. My guess is they made assumptions about you being a foreign student at a local university and took it upon themself to lecture you on proper English used in polite society, by asserting her sociolinguistic position as socially polite, linguistically correct, and thus culturally superior.

That is a textbook act of microaggression.