Early access is a much newer term, an open beta used to be a period of time you could play a beta build of a game and then give feedback to the developers on how to tweak it so they could take that information and build it out/QA it before release. Like Halo 3 before it launched, when you could apply at Bungie.net for access on Xbox live.

Then eventually the term "beta" just became synonymous with "demo" as the "beta's" started coming out within time periods that were unrealistic for the company to actually make changes to for the release date, and had no structure with which to give the developer feedback. Why? Because it's not a beta, it's a demo.

Early access was popularized after game devs began realizing that people would literally pay for a game that wasn't finished and might never be, essentially game development as a service. Thanks Star Citizen, for showing people you can make millions with a tech demo. Now broken on release and months of patches to get the game in the state it should have been on launch is the standard, because people keep buying this shit on launch day.