German Army (Heer) Breast Eagles
The German Army had a bewildering variety of insignia and the national symbol, the breast eagle, was no exception. There are two broad categories- enlisted, or issue eagles, and those privately purchased for officers. In practice, just about any combination of uniform and eagle can be found in original photos. The German troops appear to have been just as confused with the array of options and far less concerned with regulations than Living Historians.
Enlisted Eagles: During the pre-War period, most uniforms had an embroidered (woolen) eagle with a bottle green background. Sometime in the late 1930's a machine woven type (BeVo) appeared. Once the bottle green collar was dropped from the tunics, the eagles began to be manufactured with field gray backgrounds instead, and most were now of the BeVo style. A tan pattern was made for the tropical tunics, black bordered versions were made for Panzer uniforms and a simplified, triangular type was made for the M44 tunic at the end of the War.
The "M40" and "M43" are names we have given different manufacturing variations- they are NOT official German military designations. The "M43" seems to have appeared well before 1943 and is the most common eagle seen on original uniforms of all models.
Which one is correct? Any and all of them. An M36 tunic is perfectly correct with an M43 eagle and M43 tunics sometimes had M36 eagles. It's quite probable that any and all types were sewed onto any and all models of uniform at some point by some soldier. Not the answer you wanted? Sorry, the Germans were a mess. As said, the "M43" eagle is the most common. Officers often wore enlisted eagles on their uniforms.
Officer Eagles: Private tailor shops offered hand embroidered wire eagles for purchase. NCO's and Officers both tended to wear these- as well as the EM patterns. Black bordered eagles correspond to Panzer uniforms and bottle green ones for field gray tunics.