Argh, family can be hard. First, let me say that your assumption that other 'larger' personalities don't want to hang around the quiet ones is likely false. They someone to DAZZLE that isn't competing for the lime light!! You are in the power position, really. Introverts are always more powerful than extroverts. I'm one too. We get our energy from within. They need others to get it.
Family roles are hard to break out of sometimes. And they seem a little inconsiderate that they aren't 'trying' to hear you. But it is great to hear that you feel more comfortable and social in your life away from them! Keep that up. You don't have to go to everything if you don't want to with your family. :>) Pick and choose. NOT because they don't want you there. I am positive they do. But because you don't love it. So, you go occasionally and take them in smaller doses. Families don't define us. You define yourself. And you sound alright to me! Remember, WE (introverts) are powerful. They (extroverts) need us.
Ah, families. Okay, here's the thing -- families are artificial communities we don't choose. We are not only born into them whether we like the one we're in or not or they like us but we are also propagandized by religion and society to believe family is a sacred gathering. All of this is nonsense, but it's out there and we're all subject to it and it's very hard to escape. In the real world, quiet and "awkward" people are often the most attractive because they are the most actually interesting people to get to know because you can't read them easily. Easily outgoing people are easy to be with, but they also know this and that can lead to superficiality. Or not. What I'm saying is, your real family in life is your friends and the people you choose to be with and who choose to be with you. When family works it's really great but it just usually doesn't for at least someone in the family because, again, we're stuck with it, we don't choose it. So if family doesn't fulfill your needs, your friends and people you work with probably will. That's also who you will spend most of your life with, not your family, unless you're unwillingly stuck with family. But that being said, life is better when we do get along with family if it's possible. In your case, it doesn't appear anyone in your family is avoiding you. It appears, and this is just from your post, I'm not there, that you are making assumptions about what your family thinks of you and those assumptions might be completely false. When you're not there, your parents may be speaking only about you, you don't know. It might be a quirk of their personalities. You'll never really know for certain who the favorite is because what people say or joke about isn't necessarily what they actually feel. You'll never know that, assuming there is a favorite at all. People in gatherings, including family, will always stake out a territory, and that makes it a ballgame, you know? If we were all the same life would be dull. So it's possible everything you said here is in your head and not real, and that might indicate insecurity or some other mental habit that is more important to be concerned about than what family members think of you or don't think of you. You also evince some jealousy, another sign of insecurity. We all have some of it, and now you get to learn to rise above it and leave it behind. What a stroke of luck, you get to grow in a way you might not if you were that outgoing person who never had to work at it. Peace.