First some notes about your code #include "WiFi.h" is not required and #include "application.h" should rather be #include <Particle.h>.
Since you are not stating any other SYSTEM_MODE() your code will be running SYSTEM_MODE(AUTOMATIC) which renders WiFi.on() in setup() superfluous - as is Particle.process() in loop().
I also don't see the use of WiFiAccessPoint ap in your loop() as you are nowhere using it.
However, can you explain what you mean with
How do you read the date off the cloud and on which device?
Data seen via print {β1β:βAβ,β2β:βBβ}
I am using HMI Editor app on my ipad when I request data stored in Particle.variable(βWiFi_Responceβ, buf), the response is:
The publish payload is not sent as a JSON object but as string value wrapped in the publish package (which is a JSON object)
Your HMI Editor fails to correctly escape embedded double quotes inside of strings
When you use particle subscribe you will see how a string with embedded double quotes should be displayed to make clear which double quotes are constituting the boundaries of the string and which are embedded in the string
It should look something like this
It's looks like is something wrong with your HMI Editor app as @ScruffR mentioned.
I was so curious what is going wrong as I never hit this issue with JSON Buffer Writer so I made a simple html file which gets Particle.variable named json_test and flash exactly part of your code to one of my photon and I got correct answer as @ScruffR shows how this should look like: