The safe picture is the plain one: bend plus -er, the thing that bends something. A plumber's pipe bender is exactly that — you put a straight tube in, and it comes out curved. The slang meanings sit around this picture rather than clearly growing out of it: our sources call the drinking sense uncertain in origin, and say nothing at all about how the insult arose. So hold the tool in mind as the word's literal centre, and treat the other two as slang that has to be learned on its own terms.
From the word’s history
Wiktionary lists bend + -er as one hypothesis, and records the heavy-drinking sense only from 1846, calling its origin uncertain (possible links to 'bend one's elbow' or to a Scottish sense 'strong drinker'; also a sixpence, called a bender because the silver bent easily).
Wiktionary 把 bend + -er 列為其中一種說法。「狂飲」的意思最早的紀錄大約在 1846 年，來源被標成不確定，可能跟 bend one's elbow（彎手肘喝酒）有關，或跟蘇格蘭話裡「很會喝的人」有關。另外六便士硬幣也曾叫 bender，因為銀質容易用手折彎。
Wiktionary, "bender"
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