I changed the sender display name on my site, however emails received still show the original display name, why is that?
Once you send someone an email, the display name and email address will get added to the recipient's address book either manually or automatically (depending on the service you're using). Once you're in their contact list, your email address is associated with the display name in their contact list only any time they receive an email from you, or send an email back to you.
You may not always see this because sometimes email programs will hide what they're doing and only show you the display name. But since the display name is ignored by the email system, there's nothing that says the email address actually has to match the name. They can go in and enter any name in the address book to associate it with your email address (for example, John Smith).
New recipients of your emails will receive the new display name you have changed it to. But recipients that have been receiving emails from you already may already have you locked in, in their address book, with the original display name you originally contacted them with.
The only way around this is to change the email address along with the display name. This way email programs, including Gmail, will associate the new display name with the new email address.