Most BC games will actually run in One S mode on every Xbox, unless they've been enhanced.
Examples:
-Xbox Original games are all enhanced, so titles that were 480p on the original Xbox will be 1440p on Series S (basically a 3x boost on both axes). This is an improvement on One S.
-Xbox 360 enhanced titles run with their normal resolution on One S, but get a 2x boost on both axes on Series S, so a 720p title becomes 1440p. Again an improvement on One S.
-Fallout 4 will be one of the first titles to be enhanced on the new Xbox platforms and on Series S it will run at 60fps, in comparison to the 30fps on One S/X. Another improvement.
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Basically think of BC titles belonging to three categories:
1) Enhanced BC titles (original Xbox games, 360 enhanced titles and the upcoming "enhanced" games like Fallout 4): These are also improved on Series S. They might not hit the high resolution of Series X and One X, but they're still improved over the regular One S and in some cases might do some things better than One X (framerate in Fallout 4) and other things worse (typically resolution).
2) Normal BC titles: These run in what you call "One S" mode on every Xbox platform, but they might still exhibit improvements due to running on more powerful hardware (more stable framerate/better loading times etc.).
3) Xbox One games improved for One X. Titles released in the latter years to take advantage of One X. Unless changed like with Fallout 4, they will default to the regular One S mode on Series S, though the games being newer will likely also give them a higher chance of being patched/updated to avoid this. We'll see how that ends up.