STA or AP mode browser unable to connect to ESP8266-12

What is the process of 'connect to the ESP AP'

iPhone user would choose Settings -> Wifi -> connect to ESP AP

Sure you can also use the PC and choose Wifi Setting and connect to ESP AP

Aren't we doing that task as in the code snippet from post #16 above? e.g. isn't the station
binding to the ap when we use wifi.connect() and we see "mode": "sta+ap"

No, "192.168.1.0" is the Wifi-Network your PC and ESP Station is connected created by your Wifi-Router.

"192.168.4.0" is the Wifi-Network created by the ESP when AP is started

So there are two different Wifi-Networks and you can only join to one at the time with a single Wifi card.