Windows 10 has DirectX 12 "Installed" (it's integrated) as part of the Core Operating System.
DirectX 12 has support for Feature Level 9.0, 9.1, 9.2, 9.3, 10.0, 10.1, 11.0, 11.1, 11.2, 11.3, 11.4, 12.0, 12.1
The Feature Level in essence determines which Legacy Version of DirectX (well Direct3D) your Hardware can support., and as such will support Emulation.
Now for all intended purposes, DirectX 12 is still running DirectX 12... it's just Pretending to be the Legacy version of DirectX., which
most of the time works fine.
You can however still use the Redistributable to Manually Install Legacy Versions of DirectX.
This should work up to DirectX 10.0, when they stopped releasing Redistributables in favour of Integrating DirectX into the Operating System.
Legacy Windows Redistributables is an exception not a rule.
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What you have is Intel Graphics HD 3000-Series., to my recollection these are DirectX 10.1 Compliant GPU.
And when we look at the DxDiag., it lists this:
Feature Levels: 10.1,10.0,9.3,9.2,9.1
This means it supports DirectX 9.0a, 9.0b, 9.0c, 10.0 and 10.1
What's likely throwing you is that it does say:
DirectX Version: 11.1
and
DDI Version: 11.1
So, you'd assume this means it supports DirectX 11.1... but this misleading.
All this means is that it can Interface and Output via DirectX 11.1 (which is a requirement for Windows 8.0 onward)., that doesn't mean that it actually supports the Features of it.
And you can actually tell that it's something being overridden by the Drivers rather than what DirectX itself is Detecting because otherwise it should say:
DirectX Version: DirectX 11
I've provided the DxDiag from my own System for you to compare to.