Medicare is a single-payer health system for all seniors. It's a guarantee. It's administered by a federal agency with extremely low transaction costs. And people like it alot. You're covered no matter how sick you are or how bad a risk you are. And you're covered as old as you get. Universality creates efficiencies in economic terms. And it's a particular kind of generational social contract.
Getting rid of that and pushing seniors into private insurance which covers a declining portion of the money needed to buy private insurance means you're getting rid of Medicare. Most people can understand that. Pushing those people into the private insurance market and calling it "Medicare" doesn't make it Medicare.
But as Politifact counters: "All seniors would continue to be offered coverage under the proposal, and the program's budget would increase every year." That's why it's still "Medicare." This is why Republicans are spinning this issue so hard and bamboozling folks at places like Politifact.
Brian Beutler has more.