Had the exact same problem, exact same symptoms, starter would drain the battery in very little time. The starter seemed to crank relatively fast at first, but couldn't sustain it long enough to start on really cold morning. Took my mechanics a bit of time to narrow it down. Replaced the battery and starter finally, and things are golden.

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Peter
1985 300TD 4-speed 212K
1992 400E 343K
2001 E320 72K