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Wouldn't it be more like HaaS? Hardware as a service? Since they sell the terminals not just the software?

Most install the software on their own hardware. Definitely wasn't the case back in the day hence the moniker "terminal" for the desktop software.

Mostly correct, most financial and commodity institutions I've visited make heavy use of their (extremely expensive) keyboards.

Yeah, I was strictly talking about the software. Keyboard is sort of essential for operation.