I believe it might have been in Sarek's and Vulcan's interests to obfuscate somewhat, though. Perhaps it was never quite officially admitted that the fugitives were on Vulcan soil - so the issue was not an internal one of denied or disputed extradition from Vulcan to Earth or Vulcan to Federation, with the implications of disunity, but a purely external one of extradition from Federation to Klingon Empire, with some dishonesty involved as to where in the Federation the fugitives were hiding.
As for the title of "Vulcan Ambassador", we might choose anything between clinging on to the exact meaning of "Ambassador" today and deriving the structure of the UFP from there (by using "Journey to Babel" and ST4 evidence), and clinging on to the exact meaning of a "Federation" and deriving the UFP structure and the possible role of Ambassadors from there. In the latter case, we'd then probably have to accept that "Ambassador" in the 23rd century means something like "Senator" or "Secretary/Minister" in the 20th...
...Which wouldn't be that much of a leap, considering that "Minister" in the 18th-19th centuries used to mean more or less exactly the thing we now call Ambassador, right? That is, we'd have had a Minister of Vulcan in the putative 18th century Earth government, him being the human from Earth responsible for doing diplomacy with Vulcan.