Embarrassment for the person talked to

Sometimes I'm embarrassed for the person I'm talking to. Is there a better way to describe how I'm feeling? English is not my first language.

Embarrassment might not be the right term. The feeling I'm trying to invoke is the kind you feel when a student claims to understand philosophy, then calls David Hume "light reading" or "pop philosophy," despite demonstrating clear unfamiliarity with the subject. ("Hume is a bit simple and silly. He doesn't resolve important issues of existence, like why we exist," etc.)

(Just an example, of course. A similar one would be a student claiming to be a physicist after reading Michio Kaku, and then proceeding to "demolish" the Standard Model.)

I think you know the kind of person.

I want to say that I feel embarrassed for this person, but that doesn't sound right. When I'm embarrassed, it's a deeply personal thing; when I feel "embarrassed" for the kind of person I outlined above, it's sort of a disconnected experience, one with the realization of "oh dear".

I want to say that I feel something like condescension, but that doesn't sound right. Condescension is usually mixed with anger; or the definitive grouping of the target of your condescension in the category "inferior". But, again, this isn't right because the feeling is distinctly disconnected.

The word I'm looking for has a very mild "presence", if it exists.

The word "embarrassed" is used by native English speakers in exactly the sort of situation you describe. "I felt embarrassed for them" implies that you are certain the other person is doing or saying something embarrassingly wrong and that they would be embarrassed if only they knew!

Perhaps one of awkward, disconcerted, or simply bemused will be suitable.