Same agent used for Liberace, say (5,4)

I believe the answer is:

stage name

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'liberace say' is the definition.
I can't tell whether this defines the answer.

'same agent used' is the wordplay.
'used' indicates an anagram.
'same'+'agent'='sameagent'
'sameagent' with letters rearranged gives 'STAGE-NAME'.

'for' acts as a link.

Can you help me to learn more?

(Other definitions for stage name that I've seen before include "Lady Gaga, say" , "Player could go by this" , "Actor's pseudonym" , "fresh identity" .)

I've seen this clue in The Telegraph.

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