Not sure if you are still reading these OP? If it was me, I'm not sure I would be - I'm amazed at how judgemental of you and your son so many posters have been. I'm really sorry for your experience. How gutting to put lots of effort into organising a lovely party for your mum, only to have her tell you its been ruined by your DS. I really feel for you.
I think your mum behaved pretty childishly and was rude. Clearly both she and your little boy feel pretty strongly about birthdays!
For children of course birthdays are very exciting - and certainly for my DS they have always aroused strong feelings which over time (with help from adults, not disapproval) he has learned to manage. Other peoples birthdays, family gatherings etc can be hard and sometimes children's feelings lead them to behave in ways which are less than ideal. The answer is not to shout or shame them out of their feelings, but to gently help them to understand what is appropriate and what is not, and help them manage their feelings of excitement. I think your suggestion that your son could help his granny was appropriate. I wonder if your son picked up on some disapproval in your mum's comments about it being up to you to decide and this was what made him flee. I actually don't see what he should be apologising for. I don't think its appropriate to force 6 year olds to watch a present opening or sing happy birthday, or to apologise if they don't want to. His feelings had been hurt. That is not to say you couldn't acknowledge your mum's upset, and over time help your son to understand why his behaviour upset granny.
In my view an emotionally healthy adult ought to be able to share the birthday attention with their young grandchild. Clearly your mum struggles with this, and it felt really important to her that everyone was on their best behaviour for her. Personally I think she is the one with the issue here - which I imagine is deep rooted and based in her own childhood experienced.
When my son was 6 I could have written a post like yours. It has taken me three years of family therapy to recognise what was going on for me, my mum and my son, and work our how to help him develop more healthily. There is a new book by Philippa Perry (this one www.penguin.co.uk/authors/126/126019/philippa-perry.html) which I'm just reading and is a concise introduction to that world, in case you are interested.
From just this one example I don't know whether your son struggles generally a bit with regulating his emotions - as mine did and still does in more subtle ways, but if so, I would really recommend the kind of approach to parenting which that book outlines, rather than the punitive approach most posters seem to be suggesting.