As already mentioned, you never use your screen profile like that in Photoshop. Photoshop & your OS handle this part as an on-the-fly-output to the screen, not as part of your colour workflow.
Try this [I don't use windows so I'm not certain how it's applied to the desktop]
Run DisplayCal, at the end of the process it should ask if you want that set to your default profile.
This should save you having to do this in Windows itself… that's the bit I don't know about for sure, but Windows is notorious for not doing this properly, especially with dual displays. Strictly, each display should have its own profile & between them DisplayCal & Windows ought to be able to apply them correctly to each screen.*
In photoshop use sRGB, Preserve embedded, as in your 2nd image.
That means you will be working in whatever profile was already in the image. Your screen profile will be applied on the fly so you see your display accurately.
At Export, save as sRGB - you can do this with the Export to Web function. This preserves the original photo/artwork with its own profile but gives you a definite sRGB image to set as desktop.
Windows ought to then get it right.
I don't want to come across as anti-Windows, but this is actually what Macs just 'get right', with little effort from the user, across as many displays as you have connected.