How do I change the name, so my personal name does not appear when a visitor gets an email confirming their submission?

Currently, when a visitor  receives an email confirming their submission, the email lists my name <noreply@jotform.com>

I attempted to update the name in my account settings, but it does not reflect yet. Does it take a while or do I have to update elsewhere?

I presume you are referring to the autoresponder email sent to your user when they fill out your form?

You should be able to update the "Sender Name" under the "Sender" tab settings:

To learn more about the email alerts setting, please do check this guide: https://www.jotform.com/help/30-Explanation-of-Email-Notification-and-Autoresponder-Settings 

When you are testing your email alerts, do NOT use the "Test Email" button, that uses your default settings. You need to fill out the form and click the submit button to make sure that all of the information are correct.

Let us know if you are still having problems on this.