I've seen both. When I wrote the post all the docs from the gateways I had access to called it the iin with a historical note on bin. I don't have access to those docs anymore, though.

Edit: I really need to check when posting from my phone

I don't doubt you, but as an example, the recent notification of BIN/IIN prefix additions for Mastercard uses BIN 19 times and IIN 2 times.

https://www.mastercard.us/en-us/issuers/get-support/2-series...

Interesting, maybe I should update that page then.