You still need an Internet connection to boot a V2 cam, after which you do not need an Internet connection to continue to record to the SD card. After you lose power you will again need an Internet connection to boot a V2 cam. [edited to change WiFi to Internet, per @sodcam’s reminder below)
I just received my outdoor cam. I understand their operation is different, so will have to get back to you on that.
That particular Wyze help page is currently outdated and INCORRECT as far as V2 cams are concerned. It really should be updated.
For V2 cams to boot up and authenticate, you must have not just a WiFi connection to a router, but more specifically a working WiFi connection to the internet.
More details and an excellent video by Lifehackster on workarounds in the threads below.