show English and Low German both being part of the West Germanic family, but do not show English as a variety of Low German. Unless I'm misunderstanding what is meant by "variety", which seems quite likely.

I was sloppy. Sorry about that. Most of the West Germanic languages form dialect continua across the Low Lands and Germany (Switzerland and Austria). Anglo-Frisian (i.e., English, Scots, and Frisian) are distinct from Old Frankish (Old Franconian, i.e., Dutch, Flemish, Afrikaans) and Low German (lots of northern German Plattdeutsches). In fact, in the area where I lived (Bonn), the local dialects are sometimes called Middle German (between High and Low German). This area is called the Rhenish Fan because dialect isoglosses diverge there in a fan-like shape. Anglo-Frisan, Low German, Middle German, and Old Frankish all share some interesting traits that High German doesn't: retains t in words like Kölsch et, English it, High German es English eat, High German essen; 2nd Germanic Sound Shift didn't take place Eng make, Du maken HG machen, Eng two, Du twee, HG zwei, Eng apple, HG Apfel. Historically the dialects that became Old English (language of the Angles and Saxons) came from around southern Jutland and Schleswig-Holstein, where North Germanic (Danish) and Low German dialects are spoken today. The other Saxons migrated to what's now Saxony in Eastern Germany.

Ceci n'est pas un seing.