By slowing down (or occassionally speeding up) a tune by altering the playback speed of the recording device (which also thus changes the perceived key) record producers can alter the length of the product slightly. Record Producers and Disk Jockeys have a fetish about what is the 'proper length' of a piece for arcane reasons. It may also have been 'slowed down' because these people thought it was 'too fast for the market'.
If the piece is played on a 'standard D tuning' instrument, the 'proper key' may be E Minor. The E Minor key is perhaps the most easily obtained Minor key on a D Major instrument. For your guess that D minor is the proper key, then I guess that the instrument would be really playing in C Major...
But do not forget that if the musical instrument is very old, it may be pitched lower than 'standard pitch' - anywhere lower, not even full tones or semitones lower at modern 'standard pitch'.