GMnet CORE is the core of GMnet ENGINE. It is basically GMnet ENGINE without GMnet PUNCH. Back when GMnet ENGINE was called the "HappyTear Multiplayer Engine", this used to be the free version, while the other version with PUNCH was not free.
This product can not be downloaded as a standalone version.
For more information on the differences between ENGINE and CORE, and how to downgrade ENGINE to CORE, see the manual page on version differences.
GameMaker: Studio is owned by YoYoGames. GMnet is not affiliated with YoYoGames.
The GMnet logos use icons from Entypo (http://entypo.com/) and Open Iconic (https://useiconic.com/open/). They are licensed under CC BY-SA 4.0.