we know.. we know! but from where did it originate?! (he asked for the fourth time.)

From some English speakers somewhere. Snatch thief is more like cutpurse or pickpocket than purse snatcher. Compounds are very tricky and can be studied endlessly for fun. I know: you'll be asking a fifth time, so I'll go take a look at the same references who have access to.

https://en.wikipedia.org/wiki/Snatch_theft

Actually, the two examples I gave are really not parallel either. May snatch theif is a back formation of snatch theft.

Actually, it's more like sneak thief.

Searching Google Books I found its use in 1914 in a book about Pittsburgh, PA: link). Also, 1869 in Scientific American (link). Snatch cly 'a thief who snatches women's pockets' is in Grosse's Lexicon Balatronicum 1811 edition ([url=Lexicon Balatronicum]link[/url]).

Got up and left the computer and headed upstairs to my library to look at the B&M OED1. Under snatch: Shakespeare used snatcher as a synonym for thief; an 17th centuryEnglish-French dictionary had snatch pastry as a thief of pastries. Looked at Greens slang dictionary but nothing new there. Later, I'll take a look at snatch (v.) in the Middle English online dictionary. I assume it was used even then for steeling.
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Ceci n'est pas un seing.