Inspiron 1525 X3100 graphics (965 chipset) HDMI output not working

It is an interesting question that you have. You say your setup with at least one cable was working under windows 10 and the obscure debian distro. Then you say that the connection is no longer there when you reinstalled windows 10. But you have not tried reinstalling the debian distro? If you reinstall the debian distro and it still works with at least one cable, then I would post a question on a debian forum asking what is it in the distro that makes the external HDMI work, or if the connection still fails then Id assume that some hardware issue has unfortunately developed. We know for sure that the HDMI has never worked when operating with windows 7.

DrVlikhell said:

I know that the HDMI port still works because it works fine in an obscure variant of Debian Linux out of the box, with no fiddling required. I would just use Linux, but that particular variant is unusable as a media center.

Try installing Elementary OS (https://elementary.io/) -- it is a very light-weight version of Linux based on Debian. It's also a very stable, solid version of Linux. I have been using it on an old eMachines computer for a couple of months now, and I am very pleased with it.

Create the install DVD and run it as Linux Live off of the DVD without installing it. In this way, you can see if it will work without actually installing it.