When I fill out the Form 1099-SA, am I putting in my information or the information of the company that manages my HSA account

The first screen that asks for the Payer information, should match what is on the Form 1099-SA - Trustee/Payer's info should be entered.   The next screen will have you verify your information (you are or your spouse is the recipient) which most of the time will match what you entered into the Personal Information section.