Which Xbox console was the first to use the Xbox 360 name?

Xbox 360 was the first console to use the Xbox 360 name.

Microsoft launched Xbox 360 in November 2005 as the successor to the original Xbox. It was Microsoft's second home console and competed primarily with Sony's PlayStation 3 and Nintendo's Wii during the seventh generation.

The “360” branding was intended to suggest a complete, all-around entertainment experience rather than a simple numerical sequel. The system introduced a redesigned user interface, downloadable games and media, achievements, and a major expansion of Xbox Live's online features.

Several hardware revisions followed, including Xbox 360 S in 2010 and Xbox 360 E in 2013. Those names refer to redesigns within the Xbox 360 family, not separate console generations. The Xbox One succeeded Xbox 360 in 2013.