You have a web site hosted called example.com and you have a Gmail address email@example.com.
You create a forwarder firstname.lastname@example.org and forward that to email@example.com (or you set the Default Address to firstname.lastname@example.org).
You then login to your Gmail account and try to send a test message to email@example.com. But it never arrives!
It will never arrive in your Gmail inbox. It just simply won’t. Don’t try to make it.
Basically, Gmail will send the email to your server at ASO and ASO will report acceptance. Then, your server will try to forward the email to Gmail according to your forwarding set up. That’s fine. Gmail will accept the delivery and report acceptance of that email. That means that Google has accepted the responsibility for delivery of that email. However, Gmail thinks this is spam since it’s accepting mail to your email address, *from your email address* on a different server!
Bottom line: don’t test your forwarders using the Gmail account you’re forwarding to.