Agreed, which is one reason it is important in this conversation to separate false identity from trespassing.

The relevance of the security measure is that it serves to prove that the trespasser knew he wasn't allowed on the property.

Ah! Exactly! Which is why this conversation is important. The OP has said that he effectively always says no when asked if he is "Insert Name". My example is also poor, because asking someone if they are Bob Dylan also informs them that Bob Dylan is restricted from entry, which means something to the law.

A better example would be a security system based on something fleeting and baseless, like your favorite color. You might have always liked the same color, or you might change favorite colors every year, or you might just tell people a different color because no one really knows what your favorite color is but you anyway. If the security call box asked you what your favorite color was, you told them something random, and they let you in, is that illegal? The security guard never said that people whose favorite color is X are not allowed, he just asked what your favorite color was, and then he let you in.