This is the sort of thing that should be taught in history classes in school, but I honestly wonder how many schools actually touch on that nowadays.
My guess is very few. It's been about ten years since I was in high school, but I don't think we went much past WWII, even in my AP classes. I don't think we had a separate social studies/civics class past middle school and what we did learn was incredibly simplistic. We might have learned a general understanding of what Watergate was. Maybe. And we most certainly never touched on Johnson's impeachment. The closest thing to an education on impeachment that I remember getting was watching All The President's Men with my parents.
I think one of the reasons that people don't really get impeachment is because it's not well-taught in schools and the media kind of slurs it together into one thing. Frankly, an explainer on the mechanics of impeachment is not feasible within the structure of most news programs. LWT is built for the kind of in-depth explanation it requires, but most news is geared towards much shorter segments.
Plus, where LWT takes a lot of time to define terms and make sure its viewers are working from an accurate vocabulary, a lot of news shows aren't rigid about terminology. They'll talk about Nixon but they rarely make a point of emphasizing that impeachment inquires were opened but Nixon was never formally impeached. And when they talk about Clinton's impeachment in one breath and then Nixon's resignation in the next, it probably confuses a lot of people because it's a distinction that requires some time to make.