Discussing work with supervisor in an invited dinner with his family

Welcome to academia.SE :-)

See, many things that we do in our life are most of the time unplanned and spontaneous. The thing that you did is not a big deal for good people. Just say sorry sometime during tea time and get on with it. Don't keep it scratching.

Further, one positive to that is: He would have felt so happy that he has got a very dedicated (and weird) student who shows results during family dinner.

I have done it quite few times and I am not sorry about it. It feels weird initially, but not so apologetic.

Good luck!

I agree with Coder. There aren't usually any rules with informal family dinners with the supervisor. On the spectrum of interactions, it is not formal and does not hold as great a social weight as you seem to imply. It could be that he and his family really enjoyed your excitement and your passion explaining your project. It is likely that it may not be inappropriate at all. His family recognise that it is not just a "social dinner" and they know that you are his student and they are probably very used to work debate and discussions. Imagine how awkward it would be if you became overly familiar and overly social with him and his family, asking about personal sensitive issues. That would have been a far much worse scenario, with huge implications and damage to your professional relationship.

Definitely thank him and show deep appreciation for the gift, time and dinner. Take the time to figure out how best to do this, his secretary or someone who knows him might give some clues. Remember it is something thoughtful rather than the cost that matters. When people get older, "stuff" and "trinkets" are more of a burden than appreciated. At a minimum, send an email.

Don't focus on apologizing, because his might make further invitations awkward. At a push, consider saying that you feel awkward or clumsy but you were excited about your project. Gifts are not usually expected from students. Usually people bring wine or a little something for home cooked dinners from colleagues or peers but not for restaurant dinners. If it is a more formal event then the gifting follows the rules around that type of event. So well done with your project and your internship. It sounds like great work and I am sure your professor appreciated catching up.

I did not bring a gift

In most (?) cultures, a person above you in a hierarchical structure should not expect gifts from you, especially if his income is significantly higher than yours - unless he is doing you some great favor. A dinner invitation is not a great favor meriting a gift from you.

Do you think these behaviors were rude?

The only part which may have been rude is "talking shop" in the presence of his family. And even that depends on subtle aspects of your relationship with your supervisor and whether or not he hinted he might be interested in seeing the result.

It couldn't have been significantly rude, because you only did it at the end of the meeting. So, either a bit rude or not rude at all.

I was very clumsy.

Whether you had been rude or not, you weren't clumsy. You're berating yourself well beyond what your actions merit, even under the worst possible interpretation.

Could I write something as thank you letter and apologize about not being prepared?

That sounds like too much. But when you next meet, you can tell him you what you feel; you can even say you're sorry (whether that's merited or not). He will likely tell you that it's all fine and say there's nothing for you to be sorry about.

If you're not meeting him for a long time, then it's ok to send a thank-you email for the gift (assuming it's not completely trivial gift like some chocolate); and you can tell him how you feel as a secondary point in such a message - but don't over-dramatize things.

I do not know what should I do right now. I am feeling so embarrassed.

Consider some breathing exercising, or light physical activity, to reduce tension. Until you next meet your Professor.