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c.1300, "a kind of sausage: the stomach or one of the entrails of a pig, sheep, etc., stuffed with minced meat, suet, seasoning, boiled and kept till needed," perhaps from a West Germanic stem *pud- "to swell" (cf. Old English puduc "a wen," Westphalian dialect puddek "lump, pudding," Low German pudde-wurst "black pudding," English dialectal pod "belly;" also cf. pudgy).
Other possibility is the traditional one that it is from Old French boudin "sausage," from Vulgar Latin *botellinus, from Latin botellus "sausage" (change of French b- to English p- presents difficulties, but cf. purse). The modern sense had emerged by 1670, from extension to other foods boiled or steamed in a bag or sack (16c.). German pudding, French pouding, Swedish pudding, Irish putog are from English. Pudding-pie attested from 1590s.
The penis: You can't even come off unless you pull your own pudding (1719+)
any of several foods whose common characteristic is a relatively soft, spongy, and thick texture. In the United States, puddings are nearly always sweet desserts of milk or fruit juice variously flavoured and thickened with cornstarch, arrowroot, flour, tapioca, rice, bread, or eggs. The rarer savoury puddings are thickened vegetable purees, souffle-like dishes, or like corn pudding, custards. Hasty pudding is a cornmeal mush