They call each show by its name, and don't consider most Animation shows different from most other programming. Since a much larger percent of their programming is Animation and it is a much larger part of their culture.
Words like 'Cartoon' and 'Anime' were mostly created by people outside Japan. Then those terms were brought in along with the mediums over the years. Anime was used to distinguish Japanese Animation when it was brought to AMERICA and got picked up by people outside Japan AGAIN.
Still many people in Japan won't even understand what you mean when you ask them 'Did you see that great Anime last night?'
But if you just simply say 'Did you watch X show last night.' They are on the same page.
And this is why everyone on 4chan is secretly Weeaboo.