"Open source" and "closed source" are not the only options. There are plenty of products out there where you're technically allowed to look at the source code, but very restricted in how you can legally use it. The "open" in "open source" is generally understood to mean that users have permission to use, modify and redistribute the software. (Without that permission, calling it "freeware", "shared source" or "source available" would be more accurate.)

That's what "free as in beer" means -- it's a well-established phrase meaning "zero monetary cost": https://en.wikipedia.org/wiki/Gratis_versus_libre

In the case of the non-Apache-licensed version of TimescaleDB, you're allowed to use the software without payment, and you can distribute unmodified copies. But you're essentially forbidden from letting users define their own schemas, or from modifying it or reusing components unless your modified version imposes that same restriction. (The exception is if you agree to transfer ownership of your changes back to Timescale.)

Nobody's saying that Timescale can't build a non-open-source database, only that they should be clear about which parts are actually open. In my opinion, describing it on the homepage as an "open-source relational database" and then promoting it by benchmarking the proprietary version is at least a little bit misleading.

It is a phrase that Richard Stallman created, and well known in free software communities- https://www.wired.com/2006/09/free-as-in-beer/, https://www.gnu.org/philosophy/free-sw.html