The first situation: she really caught a cold. The night before, I thought it might not be serious, but I agreed to go out at your insistence. The next day, if it is serious, I won't go out. Then you should care about her and send her a medicine or something.
The second situation: she is not ill. I just don't want to go out with you. I made an excuse. I didn't want to bother you with more explanations the night before, so I gave you a perfunctory answer. The next day, I said I was very ill, so I refused you.