Wow. Just dealing with this myself. I'm on the more relaxed side myself. We don't have blanket restaurant rules for my 3 year old. If the place is quiet, sure, I'll let her walk around if she's restless. If there is a lot of traffic and servers moving around she has to stay in her seat. Or, more likely, we won't eat there.
But I have a friend who is more like your SIL. A lot like her. And she recently berated me for a lot of the issues the previous commenters mentioned. Not parenting my kids, not taking responsibility... etc.
I don't see it as a matter of pride that I'm offended. I choose to parent differently. My child will absolutely learn rules and manners and appropriate behaviors in a restaurant. But I don't expect perfection at age 3. I also avoid taking her places where she'd feel frustrated and I'd have to tell her control her beyond her tolerance. At a quiet outdoor tavern restaurant.... sheesh.
Also - giggling is a sign of stress. I don't think he was trying to be disrespectful. More afraid and uncertain of how he was supposed to respond. Forcing children to meet your gaze is very threatening to them. They meet your gaze naturally when they trust you and understand the situation.
Sounds like you handled the situation appropriately.
good luck with the future... not sure how i'm going to handle my friend...
Just thought of something else... someone mentioned that your son needs to learn to respect his Aunt. I agree with this. But the questions is HOW you are doing to teach that. I think kids can handle some pretty sophisticated rules. For instance, you can teach him that at her house we need to follow her rules, even if we don't agree with them. You can still teach him that she may not yell at him or punish him, but that he can calmly (as much as possible) come and find you if he has any difficulty with her so that you and your SIL can discuss the situation. And he will know that in her house you may agree with her, even if it is not a rule you care about personally in your own home. If you're meeting at a different place, like a restaurant, you could discuss her expectations with him ahead of time, explain to him how you care about her feelings (even if you don't) and come up with the temporary "rules" for the day. This would teach him that you care about family and their quirks, even if you don't agree with them. I've tried to explain to my 3 year old that different people have different rules and in their own house, they are the boss. I've also taught her to respect our guests - so when Grandpa comes over, for instance, she has to wear clothes at all times. Even though we don't care about naked toddlers, it bothers Grandpa :)