I don't want my purchases to be public; do I have to sign up for the University Amazon Business Account?

Yes. If you intend to use university funds, i.e. P-Card, to purchase the item for university needs, you must sign up for the University Amazon Business Account.

In addition to visibility, this account allows for stricter compliance to the university policy and ensures business-related purchases are exempt from paying sales tax.

However, only Administrators of your group and the Business Account entity will have access to your purchases.  Purchases prior to the University Amazon Business Account cannot be seen.