iCloud ID versus iTunes account

Welcome to the Apple Community.

To begin with, I would like to try to explain Apple's confusing nomenclature in regard to iCloud.

Apple have called the whole cloud thing iCloud, there are a number of features under the iCloud umbrella, some of which require their own login. iTunes is one of these, another is what Apple have unfortunately also called iCloud too.

You can use the same Apple ID (account) to login to both iCloud and iTunes, but you don't need to and often users will login to each service using a different ID.

The part that you need to remember is that the services available when you log into iCloud are completely different and unrelated to those when you log in to iTunes. Your iCloud login enables mail, contacts, calendars, find my phone, Back to My Mac, Documents & Data sharing and photostream, it does not affect any of your iTunes services.

To avoid confusion when discussing your problem, when I mention iCloud, I am referring to the services under the iCloud login, Whereas I will refer to the whole cloud thing as The Cloud.
The part that I think you might be missing, is that you cannot simply just change the ID you use for either iCloud or iTunes, you need to sign out of the accounts and then sign back in using the alternative details.

You need to sign out of your iCloud account at system preferences > iCloud and then log back in with a new ID.

Before you log out, you may wish to export your contacts and calendars so you can import the data into your new account.

That is because they are the same account, the @me address is simply an alternative address to use with your Apple ID for iCloud mail. You will need to create a new Apple ID to do what you want.

Try clicking the calendars button in the top left corner of the calendar window, to see if you had subscribed to more than one birthdays calendar. Also try unchecking any birthday calendars that you have there, to see what happens to the birthday listings in your calendar.