The ARM cores and the VideoCore parts are all in the ASIC, it's a SoC type ASIC.

Hmm, yeah it's a SoC, but not an ASIC. Maybe you mean APU? ASICs are circuits thst can only do one thing, CPU cores are definitely not that.

Edit: an example ASIC the Pi has is the video encoder/decoder, with JPEG also supported. I think it's embedded in the GPU.

ASIC just means it's an application specific IC (= chip), meaning it's fabbed for that specific product (like in this case the Raspberry Pi). A functional block like a JPEG codec contained therein is not an ASIC. Quoth wikipedia:

"Modern ASICs often include entire microprocessors, memory blocks including ROM, RAM, EEPROM, flash memory and other large building blocks. Such an ASIC is often termed a SoC (system-on-chip)."

What you're describing in the JPEG codec might be termed a fixed function IP block in semiconductor design terminology.

First paragraph of that Wikipedia article:

An application-specific integrated circuit (ASIC /ˈeɪsɪk/) is an integrated circuit (IC) chip customized for a particular use, rather than intended for general-purpose use, such as a chip designed to run in a digital voice recorder or a high-efficiency video codec.

That’s weird that Wikipedia says there’s no real distinction between SoCs and ASICS.

Colloquially, I’d never call anything with a programmable processor an ASIC

Some ASICs are SoCs. Some are not.