On the nanny word laura, I think our interpretation of a nanny is different (neither right nor wrong just different in terms of what makes a nanny a nanny as such) and like I said before it's not really the point anyway, what's in a name...
Your OP did come across as being somewhat self-centred to me and I feel sorry for this little boy who sounds quite miserable if a bit of a brat, it's true. Your last post puts across your points with a greater undertanding of the situation than is first suggested.
However I am still puzzled as to why you don't seem to know, with authority, if he can or can't watch all this TV or if you can really insist that he switches it off and does something more creative or useful instead. It doesn't sound like you know if he likes baking or lego or going to the park, it sounds like you are guessing which doesn't seem right.
I appreciate that the arrangement is such that you have a 2yo and a 6yo to entertain at the same time and that can't be easy especially if the 6yo won't cooperate well with you. It sounds like you've had a better time with him today. Maybe the way forward is to compromise like you are doing, seems a fair middle ground. After a busy day at school he may well want to zone out in front of crap on the TV and who can blame him, millions of adults do it every day instead of baking, making and running around!
I am sorry if I came across as harsh but until your last post I had a mental image of this rude but sad little boy who is having to accept a situation he doesn't like, beyond his control, and the only power he has is being cheeky/critical and watching crap TV probably because on some level he knows it bugs you, and if he's not happy, why sould you be happy.