What do you call this- ASIC or SOC?
The article you've referenced is just the tip of the iceberg. 

I think that you'll be better off ignoring all these terminology caveats about SOCs, and stick to the very simple definition of ASIC being a chip which is designed and manufactured for a specific application. If you adopt this definition, then any chip which went through all the following steps is an ASIC:


Architectural specification for a single task
Micro-architectural specification
RTL coding
Synthesis
Place and rout
GDSII files sent to fab (sometimes referred as "Tape-out")
Custom masks preparation 
Fabrication


Please note that the text marked bold in #1 is very important - it is the only part which differentiates ASICs from, say, FPGAs (FPGAs go through exactly the same steps, but their architecture and micro-architecture aims to achieve versatility over a wide range of applications).

Surely, I lied - I oversimplified the ASIC design flow and made it look like ASICs and FPGAs are almost identical, but this is the only definition I can think of which does not require too deep understanding of concepts and does not rise additional questions.

Why do I think that you should not use the term SOC? Well, it is because it is not well defined and is interpreted differently in different places/circumstances/groups/etc. 

For example: for years Intel manufactured a standalone CPUs and chips called "chipset" (chipset mainly contained the logic which "bridged" between CPU and memory/graphics/peripherals). These were two distinct chips containing digital/analog/mixed signals modules, various IOs, memories etc... Both were enormously complex. Nowadays there is a new trend - the functionality of both CPU and chipset is no longer separated, but reside on the same chip. These chips are referred to as SOCs by Intel (the latest was Bay-Trail if I'm not mistaken). Why do they call it SOC? Well, they had to give a name to a new micro-architectural approach, why not SOC? Surely, there is motivation behind this name, but this motivation is Intel specific. If you call your disk array controller a SOC in front of Intel engineers there is a good chance they will not understand you.

In summary:

Call your project ASIC (if it fulfills all the above criteria). Leave all this SOC terminology to Intel, ARM and co. to handle and use.