The search for often comes from a place of financial restriction, which is understandable. However, the game goes on sale for as little as $5 (for Rebirth) during Steam sales. Here is why buying is better than browsing:
While the base game is single-player, you can enjoy it online with friends using these methods: Binding Of Isaac Play Online
Zero lag for the host; friend only needs internet. Supports local co-op (Baby characters). Cons: Your PC must be on and running the game. The search for often comes from a place
A: You likely found the original Flash demo. The art in Rebirth is vector-based and HD. If the art looks like MS Paint, you are in the 2011 version. Supports local co-op (Baby characters)
To find it, search for "The Binding of Isaac Flash Player" on reputable archiving subreddits. You will need an actual Flash emulator (like Ruffle) or an old version of the Flash projector.