Your point here is correct. A very common tactic is for police to make a wrong claim - which is legal - such as "If you don't cooperate with me, I'm going to arrest you and charge you with X". The police are absolutely allowed to lie to you, including about the law. Likewise, they will use your reluctance to cause a fracas against you to extract easy answers.

They may even, as you say above, move to detain you and bring you in. Again, actually having a lawyer here is what's important, and again, usually not something most people can afford.