There's a blurry line sometimes, but Application-class (Cortex A* cores, like A5, A35, etc. etc.) are solidly MPU. Not even close.
The actual difference IMHO is the existence of an MMU. (Which is, fdpic/linux-nommu efforts notwithstanding, the general "Linux" condition.)
(Ex.: the SOPHGO SG2000 RISC-V chip comes with integrated DDR3 RAM, but is still solidly an MPU. I believe there are some ARM Cortex-A with integrated RAM too, can't think of any off the top of my head. [Ed.: nevermind, the SG2000 is dual RISC-V/ARM])
It’s a little outdated but this blog post is a pretty good overview of the options available [1].
[0] - https://www.nuvoton.com/products/microprocessors/arm9-mpus/
As an aside, I'm pretty sure these chips were announced months ago despite what the article says.