Companies like Google, Amazon, and Microsoft have arms for hardware design, video game development, web development, close-to-the-metal graphical programming, embedded systems, retail...that's not just big enough to drive a truck through, it's big enough for that platform they truck space shuttles around on!
You can say that those companies are unlikely to realistically claim ownership of anything that YOU didn't work closely with, but I can say from experience that these companies have explicit company-wide policies regarding some of those areas, with stricter mandatory reporting guidelines, and even requirements to use that company's technology, separate from their ordinary side project/open source contribution/etc. policies.
So I wholeheartedly agree with your second point - what you do off the clock is none of your employer's business, full stop. Just don't use company equipment for it.