If someone says '(But) I don't even feel sick.' after he is diagnosed with a disease, does it mean he doesn't even have nausea? Or, could it mean he doesn't even feel he has a disease?
In BrE, that would be "But I don't even feel ill". In BrE, "to feel sick" does mean to be experiencing nausea.
If a doctor asked me "Do you feel sick?" I would only be answering "Do you feel nauseous?" If the doctor said "Do you feel ill?" I would answer with any symptoms I was currently experiencing.
I think "sick" meaning "ill" is more AmE, but it's certainly used a lot here too these days. It should be pointed out that it has been used in one particular instance for years in the UK at least, when someone has not come into school or work because they are ill. We say "He's off sick", not "He's off ill", or "He's taken a sick day", not "He's taken an ill day".
For info, a slang term in BrE when someone phones work to say that they are ill when they're not (they just want the day off) is "to throw a sickie"!