The description you wrote maybe intend to an exercise that israel participate every year abroad,Then they have to mark them different as you described.
now i'll try to help you,first of all there is three different squadrons:
The 131 Squadron "Yellow Birds"
The 103 Squadron "Elephants"
The "International" Squadron
On most of the planes there is no markings of ISRAEL AIRFORCE and there is no israeli flag only israeli air force markings(magen david in a circled white)on the rear sides of the plane and on the tail there is the squadron markings as shown above,there are red markings on the tip of the propeller,the bottom is plane tan as shown in the pictures i posted,now probably The "International" Squadron confused you,you can add another separate skin as shown abroad at foreign exercises.
I hope this information helped you a little if not i'm still here!