As I've said many times before:
"Free Software" and "Open Source Software" refer to almost the same sets of software; but the "Free Software" movement/ideology and the "Open Source" movement/ideology are very different. And if we're talking about giving money to a foundation that does advocacy, it's the latter that matters more.
Because FS and OSS do refer to almost the same set of software, they are allies (more OSS means more FS and more FS means more OSS), but that doesn't make them friends. You are correct that OS does refer to more than just visible source. However, the OS movement/ideology does not care about user freedom as an objective, the OS ideology is based around the belief that user freedom leads to better, higher-quality software, for less dev cost; which is good business. Within the OS ideology and movement there is room to believe that there are situations where that isn't true, and that it can be the right decision to restrict user freedom. In the OS ideology, user freedom is the strategy, not the objective. In the FS ideology, user freedom is the objective; it is never the right decision to restrict user freedom.
OpenBSD is a different topic: it is free software, and considers user freedom to be an objective. However, they disagree with GNU/FSF about whether copyleft/the GPL is a good strategy for user freedom.