Although we can't actually observe what goes on inside the majority of people's heads, we can observe usage through corpora and the like, and it does seem that most people agree that the verb google means specifically "to search using Google". But it's also clear that not everyone uses the verb this way—it's generic for some speakers, and this is true regardless of what any court decides.

For every site I use regularly, I have a one- or two-letter shortcut. When I want to use Slashdot, I hit ^L to focus the URL bar, type the letters sd, and then hit enter.

If I wanted to use Slashdot with HTTPS (which I don't), I would simply change the URL associated with that shortcut to say https instead. It would be zero extra letters for me.

(Of course I don't run a secret email service.)

Exactly what someone who runs a secret email service would want us to think!