Directly via inhalation, or by physical introduction of the virus from the hands onto a mucosal membrane (eyes, nose, mouth)

Presumably it depends on the virus, but tissues of the respiratory tract, from the nose/mouth all the way down to the lungs, have many of the same receptors. Typically a cold or flu virus infects the surface tissues in your nasal passages and then replicates its way down into your lungs, assuming it cares to reach that far.

At least, that's how someone who researched rhinoviruses once explained it to me many years ago, but probably using more accurate terminology.

Also, AFAIU, these viruses have also evolved other exploits. It's no coincidence that cold viruses induce sneezing and other sinus irritations; the viral fragments themselves have evolved to trigger this behavior, which is why it can happen after a live infection is gone--your tissues are still shedding viral fragments.

That appears to be the case. That's why you ought not touch your face or your nose, and why wearing a mask is recommended.