Anyone who is fully Frum would be considered Chareidi by many people. They certainly wouldn’t be considered “Dati” by itself. “Dati” implies not fully Frum. Anyone who is fully Frum would at least be called “Dati Chazak”, if not Chareidei Leumi or Chareidi. There is a reason why the terms “Dati Chazak” and “Chareidi Leumi” exist. If “Dati” meant fully Frum, there would be no reason for anyone to call themselves “Dati Chazak” or “Chareidi Leumi”.
Before I moved into an Israeli Chareidi apartment building, the landlord asked me if I was Chareidi. He was asking about my religious level, not my hashkafos. He didn’t care about my hashakafas; he wanted to make sure that I wouldn’t do anything that would offend the neighbors.