"in nuance" means that there is a small difference in meaning. My advice is that you should not worry about such slight differences in meaning if you are learning English because your time will be better spent learning something else about English. The reality is that an English-listener will probably not even hear the difference between "I'll and I will".

Yes. "I will" has stronger meaning compared to "I'll' which sounds more casual. I'll finish this tonight VS. I will finish this tonight. It comes from the spoken emphasis.

'I will' and 'I'll' mean exactly the same thing. Usually people speak fast, so we probably say 'I'll' most of the time, but if we want to be really definite about something, we might choose 'I will' with the stress on the word 'will' to emphasise that we mean what we say. I think that might be what the teacher means.