> Why does it make sense to count these people as homeless?

It's a social class thing. It's not about having a reliable, warm, dry, safe place to sleep. It's about whether a middle class university graduate's friends would have considered them to have failed at life. Living in a coffin sized apartment with cockroaches everywhere in New York is something they could see themselves doing at some point in their life for career goals or because of bad luck. A trailer is not the kind of place a university graduate lives in.