Show your parents the Forbes college ranking and the US News and World report national liberal arts college rankings. Both rankings are available online; just Google them. Then your parents will see how amazing the reputations of Amherst and Williams are, and they will begin to understand the difference between these colleges and Stony Brook.
I live on Long Island and have taken classes at Stony Brook, having grown up right next to it. It is a fine university. But classes are huge, the campus is huge, and there are a ton of commuters (like you would be). There will certainly be some bright students at Stony Brook, including two very smart students I know who will be in Stony Brook’s Class of 2022 because it was their safety school and they were rejected at all their other options, and a third very smart student who chose Stony Brook from the beginning as his top choice. Would you be in the honors program?
In terms of your anxiety about leaving home and meeting people, you should know that colleges like Williams and Amherst often have a freshman experience designed to help all students connect with other students and feel at home. I know Williams better (I went there, and my son is going), so I will speak about it. At Williams, you will begin your experience by selecting an Ephventure orientation program, where you will spend a few days interacting with a small group of other arriving freshmen. Then, you will live for your first year in an “entry”— a dorm with 19 other freshmen and two junior advisors. This will give you an instant group of people with whom to hang out, eat dinner, etc.
The other thing about Williams and Amherst is that all their resources, including support services if you want them, are unparalleled. They take good care of their students, at no additional charge.
You mentioned an interest in science. Science is a strength of Stony Brook’s. If you want engineering, Stony Brook makes more sense. But if you want another science, note that for 25 years, Williams has had the most professors receiving National Science Foundation research grants of any liberal arts college, and, for 20 years, its alumni have won more Apker Awards for undergraduate physics research than any other college, period (as per their admissions brochures). 200 paid science research positions are offered on campus every summer— a high percentage of the 2000 students total.
I think you can get a great education at Stony Brook if you really “go for it” and make yourself known to your professors. But a student who is more shy or introverted, as you describe yourself, may actually have an easier time at Amherst or Williams. Professors there look out for their students. They will reach out to you about opportunities they think you will enjoy. It is easier to get amazing opportunities at A or W than at SB, but certainly you can do so at SB, too, with initiative on your part.
Congratulations on having such great options! You should attend Admitted Students Days at all three colleges if you can, and see which feels best. After all, you will be spending four years there, so you want to feel that your choice is right for you. Williams may be willing to pay for your travel to their Admitted Students Days, on April 23 and 24, so ask their admissions office about that if cost is an issue… although if your family has a car, it is only about a four hour drive from Suffolk County.
Good luck and congratulations again!