Do you mean on the street or at your door? All an officer generally has to do is show his badge on the street. Detectives generally have a business card and an identification card that they show when they are at someone's door. When you say is an oral claim enough, do you mean if it is shouted at you do you have to stop? The rules of when a person may or may not flee and the consequences of flight are complicated and cannot be answered in a quick manner. Suffice it to say that it rarely has an impact on a criminal case. Either the police had the authority to do what they did or they didn't - how and whether they identified themselves usually does not change the outcome.
Most NYPD officers simply show their badges ("shields" is the local term. On occasion they may also display their photo identification cards.
Your last question is pregnant with the story you are not telling us which I wonder whether it concerns an arrest.
If so, in the hypothetical situation where a police officer who has just witnessed the commission of a felony in his presence, announces himself as "police" without displaying a badge, if that's your only objection, it's still a valid arrest.
In the State of Florida a police officer who has a reasonable suspicion that someone is engaged in illegal activity can approach the person to ask for identification and to ascertain why the person is there. If the officer is in full uniform that is enough to establish his status as an officer if it relates to a later charge of resisting, obstructing or opposing an officer. If the officer is not in uniform then he would need to show a badge, agency photo identification or otherwise take such steps that a reasonable and ordinary person under the circumstances would have believed he was actually an officer and simply claiming orally to be an officer without more would generally not be enough to trigger any requirement to cooperate and would not generally be sufficient to sustain a later arrest for resisting, obstructing or opposing an officer.