It is well known that it comes from a Wade-Giles transcription of the Mandarin Chinese word for "lion dog" (獅子狗 shih1-tzu0-kou3, from 獅子 "lion" + 狗 "dog"). This is part is indubitable. There's no doubt about it.
In spite of that, there are a couple of problems.
The English Wikipedia article for Shih Tzu and some of its sources (some may have been removed) claim that the Shih Tzu is so named due to its resemblance to the lion. The Wiktionary entry for Shih Tzu even goes so far to claim that it resembles those Chinese guardian lion statues in the imperial palace or in front of feudal officials' residences.
However, the breed's name in Chinese actually translates to "Xi Shi dog", with Xi Shi being one of the four most beautiful women in Chinese history. Through the Chinese Wikipedia article for the breed and Google Translate, I've learned that this breed may have been called "lion dog" and renamed for "marketing purposes", although this claim has no sources to back it up. Meanwhile, the name "lion dog", in proper Chinese, refers to the Pekingese, a completely different breed that, if you ask me, better resembles lion statues.
So why is the English word of Chinese origin "Shih Tzu" used to refer to a dog breed not known in Chinese as "Shih Tzu"?
Was it really the original name of this breed? Or is it used in English etymologically erroneously for some reason? Was there a name change in Chinese which led to the etymological discrepancy between the English name of the breed and its Chinese name? Or was it just that the English-speaking world somehow confused two different dog breeds?
Edit: I'd like to stress that this is a question on etymology. And the etymology of English loanwords obviously has to concern with the source languages or the original circumstance around the time English borrowed from that source. I'm aware that there may not be any etymologists here who has a bit understanding of the original Chinese words ("lion", "lion dog" or the like), but unfortunately there is no "etymology stackexchange" for me to rely on.