I don't really understand why the students have less of a problem with "that" than "it". All impersonal pronouns like "this", or "those", must always refer to a thing or things previously mentioned or easily identified.
When a sentence begins with "it", it must refer to something you have already spoken about, or something apparent, such as something you are both experiencing or the conditions that you are both in.
The difference between "it" and "that" is that you would normally say "that is interesting" in response to something the other person said, or something you have just become aware of, but "it is interesting" about something you already know about or have spoken about yourself. A rule of thumb when it comes to objects is that if it is right in front of you use "it" or "this", but if you can point to it from a distance use "that". Similarly, we use an appropriate pronoun with discussion topics depending on whether we brought the topic into the discussion ourself or it it was raised by someone else.
- I have a new car. It is red.
Here, "it" refers to the car you just mentioned.
When you say "it is cold" or "it's a nice day" without previously referring to a specific place or condition, it can only refer to the current place or condition you and your audience are experiencing.
The only time it could be ambiguous is if you were referring to conditions such as the temperature and there was a difference between the temperature inside and outside. For example, if you were inside the house on a cold day and said "it's cold", it might not be completely clear if you mean cold outside or inside the house. Context would probably provide a clue, but I would assume they meant the immediate condition they were experiencing at the time they said it.