I want to address the issue with grandma. My daughter and I were at odds for several years with her saying I undermined her authority. We tried to discuss the issue and my insight was that I intervened when I thought she was being unreasonable. I could see happening with us what happened with you and the food issue. I would feel that my authority was undermined.
One view from my side is that what is happening in my home is my responsibility. I want peace and when my daughter is creating a negative atmosphere I'll intervene to try and get everyone back on a pleasant track. Of course that didn't work because she saw my efforts as undermining her. I finally realized I have to not say anything and I'd leave the room when things got tense even tho I thought if she'd trust me I could get things back on track.
In the case with the beans, I'd probably have offered the napkin too. Seems senseless to continue with that fight. I can see that you thought you were disciplining him for whining in the first place. I see it you punishing him for whining and not taking into account that he and you had succeeded in reaching a compromise which, in my view, means that there is no need to punish. Not letting him have a plate is not discipline because it doesn't teach anything. I see it as punishment which is counter productive. He came around to do what you wanted. That needs to be praised.
Your son is still only 7. He's still learning how to behave. You could teach him by suggesting that it would be better next time to not start out by whining and to ask for what he wants in a calm way. Then continue with your positive exchange. As it was you turned a positive move back into a negative situation. You created the negative energy.
You described your mother as whining right back to your son and then eventually giving in. When we whine we usually feel that we have no control. If I were you I would find a quiet time to calmly discuss how she is feeling. Based only on this post, I suspect that she's very aware of your feelings and is feeling insecure. What you do with your children at her house causes her to be on edge at the same time she knows that you won't back her.
I suggest that if the two of you could come up with rules that both of you can get behind that she'll back you and you'll back her. It's not you against her. You can partner.
Both of you, perhaps would benefit from the idea that when someone whines you can't hear them. Do not respond. Act as if they're not talking. Ignore them until they ask for what they want in a calm way. Of course, you have to teach them that you're going to do this. The first few times you'll have to say, "i can't hear you when you whine. Ask for what you want in a big boy voice?" (Do not do this with your mother when she whines. But do talk with her out of the children's hearing about how her tone of voice upsets you. I also suggest that you whine at times too. We all do.)
Both you and your mother have the same goal; to teach your children how to behave. Your mother does have a role in that. My daughter and I eventually worked our relationship out. It did take time and some difficult weeks//months before we were able to really listen to each other.
One thing that helped me was she stopped saying, "I"m their mother. You aren't!" as if I had no authority. And this came about in part because she accepted that I do have authority in my house. I did and do back her up even in my house as long as my rules are considered. An important rule is no fighting and what I saw happening is my daughter fighting with her children over unimportant things such as refusing to give him a plate.
Things my daughter would do is say in anger that they couldn't visit me again because they didn't pick up their toys by themselves. She objected to my getting on the floor and helping them. In their house she didn't help but in my house I do. I didn't see that as criticism of the way she did it but she felt that I was being critical and termed it undermining her authority when I continued to help pick up toys.
Another example perhaps related to food. At your house the kids have to clean their plates. At grandma's house they don't have to. It's a matter of different rules at different houses. If cleaning their plate is important to you then you need to discuss this with grandma and the two of you reach an agreement before the kids eat at grandma's house. Keep in mind that relationships involve give and take on both sides. Getting along with grandma means picking your battles and respecting her space and her belief's. Treat her as an adult and she'll whine less. Share the power with her. You don't always have to be in charge. And your children will be fine. In fact they'll be better off when the two of you can agree. They will see you working out your relationship and your differences in a calm co-operative manner. This is a good example for them.
My daughter and I do a better job of discussing our expectations ahead of time and out of the children's hearing. We have learned to trust each other. We've also learned to reinforce each other's rules by saying something like, "I know you can't do that at home but your Mom said it was OK to do that here." This sort of statement supports the other person.
I've heard over the years that the rules have to be the same in both houses. That just isn't true. Yes there have to be basic rules that are the same and it helps if discipline is similar but what really needs to happen is that the adults in both houses need to communicate with each other about what is important to them and work out agreements and compromises.
One very important reason that their can be different activities in the different homes is the very make up of the home. I can indulge my grandchildren because I am grandma and not mama. I'm retired. I'm not responsible for the day to day living. I have more time and more resources. My house is larger. I have a yard. I'm more relaxed. Does it make sense that I do everything the same. I don't think so. In fact, I tried doing that and could not. I am me and my daughter is she. We do things differently but as long as we respect each other and have the same goal of helping her children, my grandchildren, grow to be responsible and caring adults we can treat the children differently. We have similar values and expectations for our children. I'm not the mom but I am the grandma. Each of us has a role.