One could argue that the difference between an MCU and an MPU is whether or not it can reasonably run Linux...

It's possible, but my impression is that no MMU makes it difficult. Here is an example [1] of Linux on a Cortex M4 (a very boring STM32F4XX)

[1] https://elinux.org/STM32

Linux required an MMU at inception[1] and currently requires an MMU. Instead you have μClinux[2] which although it is integrated with the mainline kernel since 2.x, it lacks some important features, like it has neither fork() nor autogrow stacks[3].

[1] https://www.cs.cmu.edu/~awb/linux.history.html

[2] https://en.wikipedia.org/wiki/ΜClinux

[3] https://www.eetimes.com/how-uclinux-provides-mmu-less-proces...