Yes, this page will always contain the latest release of the free edition. I also personally keep the Newgrounds version up to date. All other versions may be behind.
The patron edition is currently on the same version as the free edition, but a patron-only update is coming in a few days, but any patron-only content will get added to the free edition during the next major update.
To update, just download the new version. If you’re on the same build and the same device (and the same browser for the offline browser and web versions), saves should automatically transfer. You can manually export and import save states with the “save to/load from disk” options in the save menu if necessary or to transfer systems or browsers, though.