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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 statue in the imperial palace or in front of feudal officials' residences.

However, the breed's name in Chinese actually translates "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.

Although you refer to the Pekingese as "a completely different breed", the distinction between the terms "Pekingese" and "Shih Tzu" might not have been so clear in the past. There seems to have been confusion/conflation of different Asian breeds of dogs and as far as I can tell, the history of these breeds is still not fully resolved today.

The first OED citation for "Shih Tzu", from 1921, refers to "dogs closely resembling the ‘Pekingese’ type, as also the ‘Shih-tzu’ dog and the ‘Pug’" (V. W. F. Collier Dogs China & Japan).

The second OED citation is to a Kennel Gazette article from May 1934. I haven't been able to read a copy of it yet, but the quote refers to "Apsos", so it might be the article "on the various Tibetan breeds of dogs both in this country and Tibet" that Irma Bayley reports writing. In the past, the terms "Shih Tzu" and "Lhasa Apso" were not (or at least not always) differentiated, according to the blog post "Difference Between a Shih Tzu and Lhasa Apso". That blog post also says that the Shih Tzu breed "originated from a mix of the Lhasa Apso and Pekingese dog breeds", although I haven't verified that this is true.

The "History" section of the linked Wikipedia article makes the following relevant statements:

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  DNA analysis placed the ancestors of today's Shih Tzu breed in the group of "ancient" breeds [...] Another branch coming down from the "Kitchen Midden Dog" gave rise to the Papillon and Long-haired Chihuahua and yet another "Kitchen Midden Dog" branch to the Pug and Shih Tzu.

  
  

  
  
  

  It seems that, despite the fact that you don't see much resemblance between the Shih Tzu and the Pekingese, they might be related to a certain extent. I haven't checked the sources for this statement.

  
  


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  There are various theories of the origins of today's breed. Theories relate that it stemmed from a cross between the Pekingese and Lhasa Apso, a Tibetan dog given by the Dalai Lama to Chinese emperors near the end of the 17th century.

  
  

  
  
  

  This agrees with the statement in the blog post that I found.

  
  


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  The first dogs of the breed were imported into Europe (England and Norway) in 1930 and were classified by the Kennel Club as "Apsos". The first European standard for the breed was written in England in 1935 by the Shih Tzu Club, and the dogs were categorised again as Shih Tzu.

  
  

  
  
  

  It seems like in English, the association of the term "Shih Tzu" with this specific dog breed may have started in 1934 or 1935, as part of an effort to differentiate breeds that were formerly thought of as Tibetan. I don't know where the idea that "Shih Tzu" would be a good name for these possibly Tibetan dogs came from.

  
  

I found another article that seems relevant: "Ancient Breeds – Shih Tzu, Tibetan Terrier, and Lhasa Apso", by Amy Fernandez, October 2014. Fernandez says that "in June 1934 the Kennel Gazette reported that descendants of three dogs that had been initially registered as Apsos were reclassified as Shih Tzu. [...] Shih Tzu imports continued to be AKC registered as Apso until 1952 when the Shih Tzu Club managed to stop the madness". You can tell from the word choice that this is not exactly an impartial source, but my main point in quoting this is that it seems that the Shih Tzu Club played an important role in forming the modern conception in English-speaking countries of a "Shih Tzu" breed.