No "Generate an app-specific password" section in my Apple ID
You need activate "Two-Step Verification" in order to "generate an app specific password" as shown in http://www.cnet.com/how-to/how-to-create-app-specific-passwords-for-icloud/


  For those unfamiliar, app-specific passwords are used when an app or service you're attempting to sign into doesn't support two-step verification. Instead of forcing you to enter your account password, you create a single-use app-specific password, eliminating any potential for your account to be compromised.


And as shown in your screenshot "Two-Step Verification" is not enabled...