It seems that quite a large number of cats do not receive the amount of attention, love, and interaction that they need. It varies from individual cat to cat, but there's a lot of misunderstandings among cat owners who seem oblivious to their cats needs. For example, 'zoomies' and loudly meowing in the middle of the night are signs that the cat is under-stimulated, bored, and generally dissatisfied. Meanwhile many cat owners legitimately think that all cats just do that.
You are correct that she wants attention if she comes up to you looking for pets, and you should feel grateful as that means she trusts you. A cat's trust is not a given, as their default mode of behavior is fear and mistrust since cats are partially prey animals, unlike dogs.
Cats have a hunting drive which needs to be fulfilled, so try to play with them every day if you can. The stick with a string and lure setup usually works the best, they want something that's gonna move and try to get away from them. Part of play for cats is stalking and watching, so just because they don't immediately go nuts like a kitten doesn't mean they aren't interested. You can figure out if they are in the mood to play or not usually if you pay attention. Their ears perking up is usually the first sign.
One last thing is that cats have a higher preferred temperature than humans, around 80 is ideal for them. That's one reason why they love to sit on laps and snuggle up, since we humans tend to have the house much colder than they want. So if she snuggles up to you or wants to sit on your lap, let her. You don't have to pet her constantly, just let her be sometimes and she'll be happy, and let you know by purring.
