I'm not British but the question you raised interested me. Therefore, I surfed the Net to find some answers written by native speakers and as a teacher I added my veiwpoint.Would a British member of this hallowed forum be able to give me a considered opinion?
This link may help you
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As far as I'm concrned, the word QUOTE is a back clipping. What's a back clipping?
Back clipping or apocopation is the most common type of clipping, in which the beginning is retained. The unclipped original may be either a simple or a composite. Examples are: ad (advertisement), cable (cablegram), doc (doctor), exam (examination), fax (facsimile), gas (gasoline), gym (gymnastics, gymnasium), memo (memorandum), mutt (muttonhead), pub (public house), pop (popular music).
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I hope I helped you!