I agree with the assessment that tires and suspension condition, including alignment are the most probable culprits. My 1991 350SD has its share of problems, but traction in snow and wet are nothing short of excellent. In fact, as stated by someone above, the benefits of a limited slip differential in such conditions is generally negligible.
Suspension alignment, when not in accordance with factory specifications, has the tires skidding slightly during the normal course of moving the car. Under these conditions if you add a little lubricant, like water, whatever the tire brand, the traction capability potential is seriously compromised. Ride with under the minimum pressure in the tire and you lower the threshold to losing traction even further, especially when the sidewall is rolled over and suddenly becomes the part of the tire with the highest normal force from the road.
I typically run my tires (Michelin Pilot XGT H4) significantly above the minimum tire pressures on the fuel filler flap just because I found the car understeered more than I liked with the pressures set according to the guide minimums. While there is a noted increase in harshness over rough surfaces, the steering response is better and the understeer, although still there, becomes less apparent during "normal aggressive" driving. Like I said though, I have never had this car unexpectedly transition from its understeering attitude to an uncontrolled oversteering condition in any weather and have driven it in all kinds.
1986 Euro 190E 2.3-16 (291,000 miles),
1998 E300D TurboDiesel, 231,000 miles -purchased with 45,000,
1988 300E 5-speed 252,000 miles,
1983 240D 4-speed, purchased w/136,000, now with 222,000 miles.
2009 ML320CDI Bluetec, 89,000 miles
1971 220D (250,000 miles plus, sold to father-in-law),
1975 240D (245,000 miles - died of body rot),
1991 350SD (176,560 miles, weakest Benz I have owned),
1999 C230 Sport (45,400 miles),
1982 240D (321,000 miles, put to sleep)