This is one of those weird naming conventions that many people don't really understand due to the mixing of Western and Eastern usage without really standardizing between the two.
Let's first look at some relatively popular and modern name changes in China that have now been accepted as a general standard in the west:
1. Peking has now become Beijing
2. Canton => Guangzhou
3. Chungking => Chongqing
It's noticeable that all the (former) names on the left are non-standardized transliterations of the pronunciation by the travellers who popularized the words in the west. On the right are the modern names as written in Chinese Pinyin. As a former language teacher, I'll not digress and just focus on the goal here by summarizing that China would much prefer standardizing modern usage to Chinese Pinyin which also happens to help international communication overall. I would agree with this sentiment whilst also adding that historical usage should be properly archived and referenced where appropriate.
How does this relate to Puer and Pu-erh? Well the Chinese 普洱 is transliterated in Wade-Giles as Pu-erh while in Chinese Pinyin it is Puer. Note that Wade-Giles was mainly a system that tranliterated Cantonese, not Putonghua, which is curious as we rarely use the transliteration of Bolay or its bastardized variations.
As Puer tea is a Chinese product, and Pinyin the most standard modern tranliteration of Chinese, it fair to assume that the best standardization in current times is Puer rather than Puerh. Now, as a Brit, the added 'h's and 'u's in language add history and flavour to the language but the Chinese in me also sees more value in effective written communication, especially when it comes to using written language in the form of #hashtags and #websearches.
Do you agree or do you think we should continue to use two different terms? Add your comments below and share your thoughts.