JotForm branding is displaying on the email response.

However, even though I have the Bronze Package that should eliminate the word Jotform from my visitors email response, it doesn't. I am also seeing the same thing happen with my email copy as well.

Are you referring to the JotForm Branding at the bottom of the Autoresponder email? If yes, please go to Settings > Emails > Autoresponder and delete the branding at the bottom of the email template.

I have the (BRONZE) Package of JotForm, however, your Jotform name still appears on my user response email, etc. we have our own brand name, and wouldn't want our customers to get confuse,

As I mentioned above, you can remove the branding from the Autoresponder email and also un-check the JotForm Branding option in your account settings page.

Apart from that, are you referring to the noreply@jotform.com in the notification and autoresponder email? If yes, you can change the default sender email and use your own email address as verified or SMTP emails.

Here are the guides: How-to-Setup-SMTP-for-a-Form

How-to-Add-a-Custom-Sender-Address-to-an-Email-Alert

In case, if you are referring to any other, please explain to us with the help of a screenshot so we can understand better.