It's people using "cloud" for just about anything that creates confusion. That cloud gaming you describe is closer to "gaming as a service" rather than it is to cloud computing.
A SaaS offering, like your gaming example, may or may not be based on cloud computing; but that distinction is not something the end user can see.
Cloud computing is pay-per-use and elastic, just like electricity. Cloud gaming is neither - you pay for a service, it's SaaS; maybe you can have in-game purchase, but from a resource perspective you don't pay more or less based on what you do, and you can't get things like burstable performance.
Time-sharing on a mainframe can be pay-per-use, it depends on the service provider, but it's not elastic, you can't get more resources on demand. And from an architecture perspective, it's not cloud computing because it's not distributed and resource allocation is not automated.
When you say that dumb terminals and mainframe computers in the 50s were cloud computing, not only are you wrong, you are also insulting the incredible work that took place at Amazon and other companies to make computing a convenient commodity. Next time you sit in your sofa to watch a movie on Netflix remember that this type of service at that price would have never been possible with a mainframe architecture. Your Netflix subscription is SaaS - but Netflix's infrastructure is running on Amazon EC2 and that's pure cloud computing.