First of all, Mr X's refusal is in no way the end of the interaction, nor of your charges. If your report of Mr X's actions gives the police probable cause, they can arrest Mr X, even if he refuses to identify, and even if they do not know his name. The only difference is that if they do not know his name, they cannot use his record, if any, in deciding whether to arrest him. If they do arrest him, they can and usually will search him. If he carries ID, they will then know his name. Even if he doesn't, he can be lawfully required to provide his legal name once he has been arrested.
End of it. End of my charges.
is not at all correct.
Now let us look at the actual NH laws involved. Wikipedia links to two provisions: Section 644:6 and Section 594:2. What do they actually say?
Section 644:6 provides that:
644:6 Loitering or Prowling. –
I. A person commits a violation if he knowingly appears at a place, or at a time, under circumstances that warrant alarm for the safety of persons or property in the vicinity. Circumstances which may be considered in determining whether such alarm is warranted include, but are not limited to, when the actor:
(a) Takes flight upon appearance of a law enforcement official or upon questioning by such an official.
(b) Manifestly endeavors to conceal himself or any object.
(c) Has in his possession tools or other property which would lead a reasonable person to believe a crime was about to be perpetrated.
(d) Examines entrances to a structure which the actor has no authority or legitimate purpose to enter.
II. Prior to any arrest under this section, unless flight or other circumstances make it impossible, a law enforcement official shall afford the actor the opportunity to dispel any alarm which would otherwise be warranted, by requesting him to identify himself and give an account for his presence and conduct. Failure to identify or account for oneself, absent other circumstances, however, shall not be grounds for arrest.
III. No person shall be convicted under this section if the law enforcement official did not comply with paragraph II or if it appears at trial that the explanation he gave of his conduct and purposes was true and, if believed by the law enforcement official at the time, would have dispelled the alarm. In such cases, any record of the arrest made under authority of paragraph I shall be expunged.
IV. In this section, "entrances" means any part of a structure through which entry or egress could be made.
Note first of all that Section 644:6 only applies when the person accused has appeared "under circumstances that warrant alarm for the safety of persons or property" more or less when the person has given a reasonable impression that s/he might be going to break in or commit some similar crime. "Loitering with intent" it is called in some jurisdictions.
In those circumstances, a LEO must offer the accused a chance to explain his or her purpose to help dispel suspicion. That would include giving his or her name. The accused is under no obligation to give a name, or show ID. The only penalty for not doing so is that suspicion will not be dispelled, and if the officer thinks fit, the accused may be arrested. This section might well apply to the scenario in the question.
Section 594:2 provides that:
594:2 Questioning and Detaining Suspects. – A peace officer may stop any person whom the officer has reason to suspect is committing, has committed, or is about to commit a crime. An officer may request the person's name and address, but the officer shall not arrest the person based solely on the person's refusal to provide such information.
This also applies only when an officer has "reason to suspect" the accused. The section permits the officer tho "request" (which the officer could probably do even if this section had not been passed). But it does not impose any duty on the accused to respond, nor impose any penalty for not responding. Again the only penalty is the failure to dispel any suspicion in the officer's mind. The officer may in any case act on any reasonable suspicion or probable cause that may appear. This section might also apply to the situation in the question.
Neither section really gives an officer any power or authority the officer would not otherwise have. Both authorize the officer to request name and other identifying information. Neither makes it an offense to refuse to provide such information. Neither section describes what the officer may do as a "DEMAND". Whether either actually constitutes a "stop and identify" statue might be debated, but the statute itself is what matters, not the label attached to it.
In the situation described inn the question, an officer might well request Mr X to identify himself, and explain what he is doing and why. The officer can take Mr X's response, if any, into account in deciding whether to detain Mr X for investigation, arrest him, warn him, or take other action, or take none. That is true whatever response Mr X may make, or if he ignores any request.
So these sections will not greatly change what might happen, one way or another, in such a situation.