What makes MB terminology confusing is that MB changed their model designations in the mid 90's. Often people will call a W126 cars (ex. 420 SEL) a "S" class, yet the name "S" class wasn't used when the 126 series was being made. Since your mother is talking about SD and SDL, that would be in the 126 series. Simply put any MB with a "S" in it's model name, is a large car. The "L" indicates that it is a stretched version of the forementioned large car. As someone mentioned before, an "L" was usually 4 inches longer than a "standard" "S" car. "D" is obviously for diesel.
Often someone selling their 300D Turbo Diesel will advertise it as a 300DTD or just 300TD. MB did make 300TDs that were not turbo diesels, but starting in 1981 they became 300TD Turbo Diesels. Oddlly in 1987 when MB made the W124 series 300TD, it had a turbo diesel engine, but the name of the car was simply 300TD Turbo.

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1999 MB SL500 (110,000 mi)
2004 Volvo V70 2.5T (220,000 mi)
2014 Tesla Model S 85 (136,000 mi)
MBCA member