Cutting remark in pub, third from publican (4)

I believe the answer is:

barb

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'cutting remark' is the definition.
(I've seen this before)

'pub third from publican' is the wordplay.
'pub' becomes 'bar' (I've seen this before).
'third from publican' becomes 'b' (I can't justify this - if you can you should believe this answer much more).
'bar'+'b'='BARB'

'in' acts as a link.

Can you help me to learn more?

(Other definitions for barb that I've seen before include "Arrowhead projection" , "Aggressive remark directed at a person" , "Cruel remark" , "Jibe" , "nasty comment" .)

I've seen this clue in The Telegraph.

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