Here is one definition of a general-purpose processor: if it can run any algorithm, then it is general purpose. This is not a particularly interesting definition, because it ignores the performance aspect that has been the driving goal for most processor development.
Well, I'm sorry you don't find the definition interesting, but that doesn't mean you can redefine it however you want.
It's therefore not enough for a processor to be Turing complete in order to be classified as general purpose; it must be able to run all programs efficiently.
I assume there's a name for a logical fallacy where you redefine terms in order to make your point.
With this in mind, let's explore what people really mean when they refer to a general-purpose processor: the specific category of workloads that these devices are optimized for and what those optimizations are.
That's not what I mean when I refer to a general-purpose processor.
Efficient designs in such a world will require admitting that there is no one-size-fits-all processor design and that there is a large spectrum, with different trade-offs at different points.
I didn't realise anyone was denying this.