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My personal fav is draw.io, which meets the most of the above criteria There is free offline version: https://drawio-app.com/blog/use-draw-io-offline/ and online too: https://app.diagrams.net/?src=about

A fully text-based diagram tool might solve all of the above problems nicely, but I’m wary that it might not give sufficient control (willing to be convinced otherwise if someone has had good experience with one).

UML and Graphdot are fairly powerful, with lots of tools from the academic community around them. That said they're kinda a pain to work with non-programatically

I just tried draw.io (via the web UI) and it seems pretty easy to use. The raw files can be saved as XML, which is amenable to be checked into source control, and we can also export as PNG or SVG… seems attractive so far.

PlantUML is decent as well.