Edge there means that the either HTTP header or META (don't remember which one exactly) won't specify IE version number and instead will use whatever is the latest version of the browser is. In IE10 this will mean IE10's engine, in IE11 this will be IE11's. This convention is several years old and predates Edge the browser.

I don't think "Edge Mode" rendering in IE11 has any relationship to the browser also named Edge. "Edge Mode" was around at least as far back as IE8. It's not the proper name of the engine.

The problem is that you have all your corporate customers who still insist on using webapps written with ActiveX and webpages that are still accessed by IE6 users. They would be more than a little upset if Microsoft told them "we're not supporting you anymore".

Microsoft is stuck here: they want to move on to modern web standards, but they can't afford to piss off their corporate customers.

Their solution is to split the web browser, shipping Edge for the consumer market (and the modern members of the business market) that wants to interact with the modern web, but still continuing to provide IE11 along with the OS to keep their corporate dinosaur customers happy.