Yes, and this one is completely legitimate. Mojang keeps the original version online for free.
Minecraft Classic, the original 2009 build of the game, is hosted by Mojang at classic.minecraft.net and runs in a browser. It is free, it is official, and it is genuinely the game as it existed before survival mode and crafting.
This is the best-case version of browser gaming: the rights holder deciding to give it away, so nobody has to worry about legality.
The full modern game does not run in a browser, and any site claiming otherwise is not telling you the truth.