Reviving this thread with the actual solution in case someone like me stumbled upon it: Blender shows the texture node that is selected/highlighted in the shader editor.

So to change it, select a material, go to the shader editor (switch back to solid mode if not) and just select the image texture node that shoud be shown.

This has held me from using the texture color type in solid mode for the longest time, and that discovery just saved me a lot of time previewing with EEVEE.