And if we're both using generally defined PHP exceptions, like InvalidArgumentException or BadMethodCallException?

Well I'd strongly recommend you to start using your own exceptions.

Currently if you're using the same exceptions, and you don't want to catch some of them, while they still have the same type, I don't know how you can reason about such source code. "Hey, here's code that throws BadMethodCallException. Sometimes I want to catch it and sometimes I don't."

You can use techniques like stub files https://phpstan.org/user-guide/stub-files or dynamic throw type extensions https://phpstan.org/developing-extensions/dynamic-throw-type-extensions to suppress vendor source code documentation about @throws, but I wouldn't recommend it.

I don't know how you can reason about such source code

Yes, normally I should either catch all of them, or at least document the ones I want to let bubble up. But if you're using a third party framework properly, or even a method from your own code, then the exceptions thrown by the framework or that other method will never actually get fired, so there's no point in catching or documenting those. I would have rather seen the purpose of the missingCheckedExceptionInThrows setting as making sure you document your code properly, and if your method/function throws an exception, then that should be documented above it.

I would at least add a separate config item for this case, if possible.