This logo is used for Xbox Live on original Xbox consoles. This logo went out of use on April 15th, 2010.
This logo is used for Xbox Live on Xbox 360 consoles.
This logo is used for Xbox Live on Windows Phone.
The font for the "Live" word was changed in 2013. This logo is used on Xbox One consoles, Windows and later Xbox 360 consoles, and soon, Xbox Series X|S. Xbox Live is also on Android, iOS, and Switch.