I know what the law says on this. I don't live in New York state but I was one of the victims of an "armed robbery" years ago. The woman passed us a note demanding what she wanted and it said she had a gun. Of course, we gave her what she wanted.
I don't truly believe she ever had a gun and none of us actually saw one. There was never a gun seen on video of the incident and no gun was ever found when she was later apprehended. The note, however, was enough to convict her of armed robbery. It's the intent to make the victim believe they are armed and not always if they are actually armed. You're not saying anything here that has any application.