Is it possible to login to Xbox Live and appear offline, so that my friends and others looking at my profile wouldn’t get the notification or know that I am online?
The answer is yes.
To appear offline on xbox live take the following steps:
Press the guide button on the controller, chose personal settings, chose online status, chose appear offline.
This will need to be done each time you turn off your console and for it to work you will also need to turn auto sign in off, otherwise your xbox 360 will sign you in when you turn it on. To do this turn on your xbox 360 without it connected to the internet first, that way your console will be unable to find a connection to log you in online.
You can also change your settings on xbox.com so that you only appear online to certain groups, such as friends rather than appearing online to all which may be useful if you dont want to get messages from none friends after playing them online as you will appear offline.