Canine tooth one has right in middle (5)

I believe the answer is:

corgi

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'canine' is the definition.
(I know that corgi is a type of Dog)

'tooth one has right in middle' is the wordplay.
'tooth' becomes 'cog' (cog is a kind of tooth).
'one' becomes 'i' (Roman numeral).
'has right' becomes 'r' (I can't justify this - if you can you should give a lot more credence to this answer).
'in middle' means one lot of letters goes inside another.
'cog'+'i'='cogi'
'cogi' enclosing 'r' is 'CORGI'.

Can you help me to learn more?

(Other definitions for corgi that I've seen before include "Dog at Buckingham Palace" , "Queen likes this" , "Dog of Welsh breed, royal favourite" , "Small Welsh dog with fox-like head" , "Pre-2018 royal dog" .)

I've seen this clue in The Times and The Sun.

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