I understand what you say. And maybe it is because I never played catch or having a tea party with my parents (the equivalents of such activities included) I don understand the situation that good.
But I think that it is a bit more then a silent wish (can one say that in English?) but it is mixed with a layer of sadness, that might grow in the future and then burden the relationship.
You do not need to have the same interests to enjoy spending time together. You can do what one person like and next time what the other person likes.

I think the situation is like this.

A father whose hobby is watching baseball games and loved playing baseball all the time in his youth is likely to play catch with his son. This mother is just expecting the same thing to her daughter. You know, it’s always good to share wonderful time with anybody who has something in common. And it’ll be more wonderful if the somebody was your son or daughter. That’s what the mother wants I think, and it’s nothing more than that.

The mother is not trying to force something to her daughter, and she probably thinks it’s ok to be girly (as you think, it’s nothing to do with dependent or independent). I think she will love her daughter no matter how girly the daughter is, needless to say. But, you know, the mother has a slight hope that she can one day have fun together for things she enjoyed very much in her youth.

You both are correct.
The mother, who was never a girly girl, is accepting of her daughter being different from her. Yet at the same time she is hoping that as her daughter grows, she does take an interest in the ‘less girly’ things like a miter saw. (It’s also supposed to be slightly humorous, tea parties being feminine and miter saws being masculine.)

I have done a quick Internet search and came across this info: http://en.wikipedia.org/wiki/Miter_saw For me, it just sounds off topic.

Maybe she is trying to say that her daughter and she may have occasional conflicts, from time to time, but it’s ok for her daughter to have her own tastes that are very different from hers.