How the search system works
RDR 2 has an understandable legal system. If Arthur steals a horse or starts threatening someone with a weapon, he automatically comes to the attention of the police.
It is worth remembering that any passer-by who witnessed a crime will seek to report it to the cops. A mark on the map will help you distinguish a witness from an ordinary person: a small image of an eye. In that case, if the witness leaves, the police will certainly find out about the crime, and the main character will be hunted.
How to get rid of the criminal status
The first thing to do is get rid of the witness. The quickest way is to simply kill it with a precise shot in the back while it is running away. Another option is to chase him, catch up and intimidate him so that he forgets what he saw. In this way, the persecution will be over.
In that case, if the witness went and told Morgana the cops, the latter immediately goes into the status of a wanted criminal. A warning about this will appear in the upper-right corner of the screen and will light up in large red letters. The area marked in red will be displayed on the minimap. That's where the cops will be hunting for Arthur, so you shouldn't be in it.
How to remove the criminal status in RDR 2
If Arthur does get into it, he must hide from the police, who can hunt him at any time of the day. Being outside the red area, the "wanted" status will gradually disappear from the screen. As soon as he's completely gone, the cops will stop chasing him.