Greeting content changed by a third (5,8)

I believe the answer is:

happy birthday

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'greeting' is the definition.
I know nothing about this answer so I cannot tell whether it can be defined by this definition.

'content changed by a third' is the wordplay.
'content' becomes 'happy' (content can mean happy or pleased).
'changed' indicates anagramming the letters.
'by'+'a'+'third'='byathird'
'byathird' with letters rearranged gives 'birthday'.
'happy'+'birthday'='HAPPY BIRTHDAY'

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(Other definitions for happy birthday that I've seen before include "World's most popular song, they say" , "Song sung at an annual party?" , "Anniversary greeting" .)

I've seen this clue in The Telegraph.

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