My 2nd cat was like this as a kitten, very playful but not real affectionate. We too, think she's mixed with Siamese and part tabby. After we got her fixed, she really seemed to settle down, although still playful and a little rascal. But now that she's almost 2 she's much more affectionate and will come and sit on my lap. She's still not quite as affectionate as my first one but she's getting there. So, give her time. With age and getting her fixed, she'll settle down a lot and may become that lap cat after all!
Not liking being picked up or moved around is very common for cats. I have several kitties (I have my own and I foster rescue cats) and none of them like to be picked up or moved.
If I want to cuddle, I wait for them to come over - which they do. The best thing is just to pet them, be close to them... etc. Give her lots of love and attention and she will want to be near you.
Some cats never really want too much physical closeness, like sitting in your lap. Often they prefer to sit next to you and just be close.
Let her take the lead and see what she is comfortable with.
Waldo used to not like being picked up much at all. He still will look pretty grumpy when picked up, but he tolerates it to a degree. I kind of gradually picked him up and held him for longer times. When he starts struggling then I set him back down.
He also doen't like sitting in my lap. Waldo's not a lap cat, but he will cuddle next to me when he wants to sleep near me.
Also it could be how you're holding your cat. Some cats tolerate being held different ways than others. Waldo tolerates it better if I hold him so his legs are in my arms and his chest is up against my shoulder. They like to have support under their rear feet.
I have a really good natured, lovable cat (not Siamese) who doesn't like to be picked up. He will jump on my lap or sleep beside me but doesn't like to be picked up. I don't think it's weird, it's just who he is. Maybe your cat just doesn't like it either. Some cats like to be in control - who am I kidding? ALL cats ARE in control = ) If your cat makes you feel good then we have one thing in common - my cats help my anxiety and have helped me a lot with a recent illness.
Hairballs and cat treats to your cat = )
I have a cat that will not allow anyone to pick her up. She is a grey tiger stripe kitty- no type I guess. She's never allowed anyone to pick her up. I've learned at the vet to take her in the carrier that I can pop the top off and let her lay in the bottom- rub her tummy and she is in heaven- pull her out and you will be bloody.
Now my all white (unless he's been rolling in the dirt) half-siamese elephant is okay with being picked up, but doesn't cuddle that way.
Kitties just have thier own purrsonality and like what they like I guess.