Cannot use Mac Mail to send email via the two gmail accounts (e.g. iPhotos)

You can effectively turn off push notifications for a particular email account... sort of. Go to the Settings app, and hit "Mail, Contacts, Calendars". Go to "Fetch New Data", then scroll down and go to "Advanced". For the email account whose little red badges you don't want, change the setting to "Manual" instead of "Push".

This will prevent new messages from pushing a notification icon to the home screen. However, when you open the Mail app to check your other email, the app will sync the "Manual" account with the server and you'll get a badge if there are unread messages there. So, like bmike mentioned, the real way to get rid of those badges is to get closer to inbox zero.

One way to fix this is to install the official server app in order to re-enable this feature. You don't have to setup a mail server through the app. Just installing it will allow use of mailx through the command line.

In the past, you had to buy and install OS X server as a separate product and now it's available from the App Store and gets installed on top of the normal OS.