Ahhhh ha ha ha. I am laughing so hard at your question since I can relate so much. You went from young child to future complex in about 5 seconds. Pre-worry up the wazoo. That's me. I think...hmmm...if I persuade them to eat these sardines will I have to pay for therapy in 6 years for an eating disorder? Will I be remembered as "The Mom who made us eat strange stuff?"
I'm trying to make things "not a big deal" like Everyday Paleo suggests but at the same time I do talk about sugar/wheat etc. and what makes us sick. My kids DO ask for things the other kids have but I just listen, nod and ignore it. That seems to work--they forget about it. They are 5 and 6.
For the past year I changed little things here and there for them and started introducing new foods as I got more grounded in Paleo. Going to the last large assembly at school and seeing all the over-weight, puffy-faced kids made me re-double my efforts more aggressively. I just wanted to run over and hug them and give them some beef liver! Ahhh ha ha.
To answer your question....I don't think you can minimize the influence. I think it's going to be like anything else...they look to you for the boundaries. And...I think they like the boundaries. It's a safety thing maybe? Plus...I don't push perfection on them. They still eat cereal and pizza with Daddy.
I don't forbid anything. I just control their lunches and snacks and try to tell them things about food. Like..."What foods are good for us!?" They yell, "Meat and Fat!" I also find that letting them help cook dinner and make their own lunches helps. They get really excited and don't think about the food as much as the fact that THEY made it. They love to make coconut milk ice cream in this easy ice cream maker we have, for example. They love to crack eggs.
It's not easy though..my daughter will love something and suddenly say..."I don't like that anymore." In my head I think, "Ok great. I'll just put some toxic grains in your lunch then cause I'm all out of ideas!"
Lots of kids from all different backgrounds/ethnicities have to to conform food-wise in society...vegetarians, Muslim kids, Jewish kids and these parents have the same challenges. I think we just have to do the best we can and hope they adopt our philosophy.
And...you are very lucky!!! Your daughter is young and so you've got lots of time to get her taste buds going in the right direction. You're going to do great! Good luck. :)