Hi Johann,
The Albatros line of monocoque, plywood single-seater fighters, D.I, D.II, D.III, D.V, D.Va were all water cooled, Mercedes D.III 6 and 8 cylinder inline engines.
There is no relationship between the Mercedes designation of there engine with Albatros' designation of the plane.
One of the disquishing features between the early models (D.I-D.III) was the placement of the radiators (there were other differences).
The "D" (for the aircraft) comes from the German word doppledecker, (2 wings), and Idflieg's (Inspektion der Fliegertruppen) designation for a single-seat fighter.
The D.I superceded the Fokker E.III (eindecker), owning to its greater speed and manuverablity.
The D.I won back air superiority which had been lost when the French introduced the Nieuport which in turn had outclassed the German E.III.
Gosh don't get me talking about WWI aviation...I don't know when to stop.
Thanks for asking and giving me the opportunity to lecture
Richard