There are sporty introverts in plenty of chapters that would welcome you with open arms.
I honestly don’t know about rushing as a grad student, but my advice if you did or say, if you transfer as an undergrad and rush, would be to understand there are some houses that look beyond girly/confident PNMs and some that don’t. If you got recommendation letters, you could ask that they include these traits (some rec forms actually ask this—“what kind of person would she be most comfortable talking to?” etc..)
But bottom line, this is YOUR life. Your mom had her college experience. This one is yours.
You acknowledge being very different than she, so obviously your experience wouldn’t be the same. Whether you want to explore recruitment is YOUR decision. I was thrilled for my daughter when she found a house that means as much to her as mine did to me. But if she didn’t care to explore a Greek life, that would be fine.
Some questions to ask yourself—
Is this something that interests you at all?
Are you afraid of trying recruitment and feeling like you’ve failed if you don’t get a bid?
Are you more worried that you would pledge for her but not be happy in a sorority?
Do you feel that not going through recruitment is somehow thumbing your nose at your mom?
Did you choose an non-Greek school to avoid the issue?
Do you feel that you are missing out on something or more that you didn’t turn in an assignment?
You do not need to fit the mold to be a wonderful contribution to any organization. I think you need to decide if this is a monkey that needs to be removed or something you wish to explore. If it is not, that is perfectly okay. It’s your life—live it as you, not as your mom.