Nope, you have the procedures right. The reasoning is still probable cause. Evidence in hand, which I assume you mean enough evidence to present to a court to determine guilt, is probable cause. Law enforcement cannot make an arrest unless they have probable cause, otherwise the arrest is unlawful. Now this does not mean they have to have all of the evidence that they are going to present to the judge or jury, just enough evidence that all of the elements of a particular crime have been met with the evidence they have in their possession.
For example, if a police officer witnesses A hit B on the head with a steel pipe, the police office has probable cause to arrest A. The police officer acyually onserving this battery is enough evidence to convict A. But, most likely, the pipe will be admitted into evidence, any blodd evidence on that pipe, B's testimony, etc. On the other hand, let's say that the police officer arrived on the scene after B was hit, the pipe is lying on the ground, A is long gone and B informs the police officer that A hit him on the head. Does the officer have probable cause? Not yet. A is a suspect worth investigating, but there is no evidence other than B's testimony. Now once the investigation proceeds, probable cause may arise (ie, A's DNA found on pipe, another witness saw A running from the scene, A has B's blood on his clothes). But to put it simply, in order for police to arrest and charge you with a crime, they need to have evidence tending to prove each and every element of the crime.