I am a beginner singer, and during some singing classes I took the teacher showed me that when the air is coming from the diaphragm, it is a little cold. She showed me some exercises and I learned how to take out that cold air flow. But later at home, when singing, I noticed my air is not cold at all, even when I am completely sure I am using my diaphragm to take the air out. Am I doing something wrong or have some wrong info?
The "cool air" vs. "warm air" question is a valid one, but doesn't have anything to do with your diaphragm.
Diaphragmatic breathing is all about supporting your breath and airstream properly. The issue here is that when your lungs expand, they need room to do so, and expanding down into negative space created by your diaphragm will allow the diaphragm to do its job when you need to exhale. The alternative would be to expand out into your upper chest, where the musculature is not designed to move air in and out of your body. I disagree that you cannot control your diaphragm -- it may not be a directly controlled muscle, but I can certainly feel and influence where my air is going when I inhale--and this takes practice.
When a voice teacher is talking about cool air or warm air, they are probably (at least if they have any hope of making sense) talking about your oral cavity.
Try breathing on your hands as if to warm them up. Now, try blowing across across a spoon of hot soup to cool it down. Bingo: instant warm air/cool air. Now, do the same while paying attention to the shape of your oral cavity, in particular with regard to the placement of your soft palate. You should notice that the soft palate is very high when blowing "cool air". That is the point of this exercise in the context of training a voice--to raise the soft palate.
You can't control you're diaphragm; it moves involuntarily. Your teacher has given you incorrect information, which I will outline and correct below.
Also, air cannot "come from the diaphragm". That is physiologically impossible. Your air comes from your lungs; nowhere else. Your diaphragm is a muscular membrane that sits below your lungs and above your stomach. When you inhale, the diaphragm is pulled downward creating negative pressure in your lungs, causing them to expand and intake air. The opposite process happens for exhaling. So, all the air that you have comes from your lungs. You can't choose where it comes from.
Quality of tone production depends on how well your vocal cords resonate. Cold vocal cords do not resonate just like cold hands make it hard to play an instrument. Therefore passing cold air through your vocal cords is actually the opposite of what you want. This is one reason why people say, "I need to go warm up."