How to show that my Internet connection is still up and active in menu bar?
With regard to modifying the system AirPort icon, I do not believe that what you are asking for is possible.

To show internet connectivity, I use a combination of WiFi Signal (which could be substituted for the OS X AirPort menu bar item) and Cloud:


AirPort off


AirPort on &amp; not connected


AirPort connected &amp; no internet


AirPort connected &amp; internet




This works very well and takes up no more space on my menu bar as I use Cloud anyway.

Dropbox is also a good alternative as it also shows network connectivity, although it is slightly slower to respond especially on slower connections where the network test takes longer.