IOS – How Apple.com set Apple Touch Icon

Bingo! you've already got it there in your question.

Your last item "Web clips Credentials and keys" should actually be two - "Web Clips" and "Credentials and Keys" in which case, "Web Clips" is the one you're looking for. A "Web Clip" is a home screen bookmark for a website.

From the Enterprise Deployment Guidelines:

Web Clip Settings

Use this payload to add web clips to the Home screen of the user’s device. Web clips provide fast access to favorite web pages. Make sure the URL you enter includes the prefix http:// or https://—this is required for the web clip to function correctly. For example, to add the online version of the iPhone User Guide to the Home screen, specify the web clip URL: http://help.apple.com/iphone/ To add a custom icon, select a graphic file in gif, jpeg, or png format, 59 x 60 pixels in size. The image is automatically scaled and cropped to fit, and converted to png format if necessary.

On iOS, HTML5's local storage is not available in private browsing mode, which means you can detect it by attempting to save something to local storage and catching the exception with the following JavaScript:

try { localStorage.test = 2; } catch (e) {
  alert('You are in Private Browsing mode');
}

^{Modified from https://stackoverflow.com/a/17741714/}

On OS X, there's not really a canonical method, but this should provide some useful information:

• Can web sites detect whether you are using private browsing mode? –Security.SE

If you're looking for implementation guidance, you should probably try Stack Overflow instead