The General Rule: The Prosecutor And Not The Crime Victim Decides
You are correct that this is mostly wrong.
Pressing charges is something that happens. But, this simply consists of making a report to law enforcement about an incident and asking that the prosecutor either directly, or indirectly through a request directed to the law enforcement officer to whom a report is made, to pursue criminal charges in the case.
But, this request has no binding or legal effect, even though many law enforcement agencies and prosecutors offices have a formal or informal unwritten policy of honoring a crime reporting victim's wishes whenever possible. In the United States, a prosecutor can commence a criminal case against someone without consulting a victim of a crime, or over the objections of the victim of the crime, and can even (and sometimes does) incarcerate the victim of the crime to force the victim to testify against a criminal defendant (this most often happens in cases of statutory rape, domestic violence and cases involving organized crime or gangs).
Likewise, a prosecutor (in the United States) can choose to decline to prosecute a criminal case despite irrefutable evidence tied up with a bow and the pleas of the victim and lots of other members of the general public.
(Not all countries are like this. For example, in Germany, some crimes can only be prosecuted with the victim's consent, and within reason a prosecutor is required to prosecute every case in which there is clear evidence that a crime was committed and the victim's consent, if necessary, is present. While Germany is something of an extreme case, it is much closer to the norm than the U.S. is in countries that do not have a common law legal tradition, while the U.S., while an extreme case, is closer to the norm of countries in the common law tradition of British law.)
A crime victim's wishes aren't entirely irrelevant in practice. Many states have a "victim's bill of rights" that requires a prosecutor to confer with a victim when possible over whether a criminal case will be commenced against a defendant, and in practice, many prosecutors in ordinary type cases come to think of the victims of crimes, rather than "the People" as their client, even though that isn't technically correct from a legal perspective.
Most of the time, a crime victim's wishes will be respected by prosecutors whether conveyed directly to a prosecutor, or indirectly through a law enforcement officer. But, nothing requires either a law enforcement officer or a prosecutor to do so.
Law enforcement officers in some ways have even more discretion than prosecutors do and simply don't pass on information about myriad crimes that they have knowledge of and could prove, every day. But, prosecutors can choose to disagree with a law enforcement officer's exercise of discretion when they do know about an incident, either by declining to prosecute a crime, or by prosecuting a crime that law enforcement officers didn't want to prosecute. And, law enforcement officers as a matter of practical reality can be forced by prosecutors to execute search warrants and arrest warrants.
Also, it is true that when neither law enforcement or prosecutors know a crime has been committed that they can't prosecute it, and usually, the way that they learn of a crime is that a crime victim reports the crime to law enforcement. The decision to report or not report a crime not known to law enforcement is, in practice, what "pressing charges" or "not pressing charges" usually means and this does, in practice, constitute a veto power over further criminal proceedings arising from an incident, even though the language used implies a more formal step in a legal process than it does.
Exceptions To The General Rule
N.B. Notwithstanding the foregoing, there are a few exceptions to the general rule.
A handful of U.S. states, so far as I know, all on the East Coast (and also England and South Africa), allow individuals who are crime victims to directly commence the prosecution of minor crimes in a court of law without the intervention of law enforcement or a prosecutor, or grand jury, often without an attorney. Usually, if this is done, the prosecuting attorney's office can take over the case from the private individual litigant if they choose to do so. A discussion of the historical practice with case citations can be found here.
In the roughly half of U.S. states and in the federal system, where felony criminal charges must be brought via a grand jury indictment before they can be tried, a so called "runaway grand jury" can bring criminal charges that the prosecutor has a duty to make a good faith effort to prosecute, without the request of the prosecutor convening the grand jury.
For example, in a recent Alabama case, prosecutors went to a grand jury seeking felony charges against someone who shot a pregnant woman, and the grand jury decided to indict the woman who was shot for manslaughter on the theory that she caused the death of her unborn child by starting a fight that led to the shooting. The prosecutor's office once it had control of the case, and had observed the public outrage over the decision, then dismissed those charges.
There are a few states where a crime victim who is dissatisfied with a prosecutor or law enforcement officer's exercise of discretion can apply to a court to have a disinterested and uninvolved attorney appointed as a special prosecutor to review the facts of an alleged crime and decide independently regarding whether criminal charges should be pressed in a case where law enforcement and the prosecutor's office declined to do so. This is most frequently invoked when the alleged perpetrator of the crime is connected to law enforcement or the prosecutor's office.
Also, just become a prosecutor declines to bring a criminal case despite a crime victim's desire to "press charges", this doesn't mean that the victim can't sue the person who committed the crime for money damages (including punitive damages in many cases) from the same conduct that constituted the crime. In big dollar cases involving fraud or theft from a small number of affluent victims (a kind of case I frequently handle in my law practice), this happens all the time.