"The report doesn't say that he refused to identify himself. It says that he refused to show a tangible piece of ID."
and that is failing to identify yourself. It was when I worked the streets in uniform and I'm sure it hasn't changed since then. Verbally stating who you are may be accepted as identification but the officers are well within their rights to ask for a form of identification to verify that which may have been spoken verbally.
"Among the circumstances which may be considered in determining whether alarm is warranted is the fact that the person takes flight upon the appearance of a law enforcement officer, refuses to identify himself,"
Refuses to identify seems pretty clear to me which according to the above warrants "alarm".
When you are asked to identify yourself at town hall, or the registry of motor vehicles, or any place else, is giving your name verbally accepted? No, it's not, they are asking for a form of identification when that request is made.
"Either way "a man with a gun" is not a crime any more than "a man with a car." What were they investigating?"
Perhaps his motive for attempting entry into the building? His mindset and overall cognitive abilities [ to determine if he might be on meds, drunk, mentally impaired, etc which are all real possibles based on prior work in the field by officers ].
Their response starts an field investigation, first with the complainant who called, maybe witnesses at the scene and then the subject of the complaint.
They write a report based on their investigation and field inquiries. Again, the police didn't just stop the man on a public way, they were called and responded to a complaint. Well within their rights to determine his state of mind based on some articulable action that caused someone to be alarmed and call the police.