Which Microsoft Store predecessor shipped with Windows 8?

The Microsoft Store's predecessor that shipped with Windows 8 was called the Windows Store.

Microsoft introduced the Windows Store as an app-distribution service alongside Windows 8. It was designed especially for Microsoft's new “Metro-style” apps, which ran in a more controlled, sandboxed environment and followed Microsoft's certification requirements. The service was announced at the BUILD developer conference in September 2011 and became available during the Windows 8 Consumer Preview in February 2012.

Windows 8 itself launched on October 26, 2012. At that time, the product was branded Windows Store, not Microsoft Store. The name changed after Microsoft unified several storefronts, including Windows Marketplace, Windows Phone Store, Xbox Music, Xbox Video, and Xbox Store, into the Microsoft Store in Windows 10.

Windows Store is therefore not merely a generic description: it was the official earlier name of the platform. Windows Market, Software Gallery, and App Center were not the Microsoft product names for this Windows 8 service.