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 was used on Xbox One and Xbox Series X & S consoles, and Windows. The logo is still used on later Xbox 360 models, and the name is still used on the Nintendo Switch, Android and iOS.
On March 21, 2021, two days after the name got leaked, Microsoft confirmed that the Xbox Live brand would be renamed to simply Xbox Network. This was to separate the online service from the Xbox Live Gold membership name. Since that's the case, Xbox Live Gold will keep the "Live" branding.