That account information is not used by the email/messaging application - as far as email is concerned, it is only used by the GMail app. See below.
If you do not want mail delivered to the GMail app, tap the gmail icon in settings->accounts and uncheck "sync Gmail". That's your first step.
Second, back in settings->accounts, ignore the google account that's there - you have just turned off gmail sync anyway, and, if you looked, is also used for other push syncing for calendars, contacts, etc.
Tap add account at the bottom, choose email from the next screen, and enter your gmail account information. That should set up your Gmail account in the email app.