Bitcoin Core is the node software that grew out of Satoshi's original 2009 client. Most public full nodes run it. It checks every block and transaction against the rules, and it includes a wallet.
How it is made
The code is public under the MIT License. Anyone can propose changes. Changes are merged only after public review. No one can force users to upgrade: each node operator chooses which version to run.
Verify, don't trust
Releases come with SHA256SUMS files signed by several builders, and are made with Guix reproducible builds. Several people can rebuild the exact same file and confirm that it matches.
Takeaway: Bitcoin Core is the most used implementation, but the rules are whatever users choose to run.