If you were using ‘dragonfly’ in a Mandarin brandname (traditional products for hair & skin), would you use 蜻蜓 or only 蜻?
In general, shorter 2-syllable brand names are better and apparently 蜻 can be used for 蜻蜓:
For example, 'Dragonfly Valley' could be either 蜻蜓谷 or 蜻谷:
https://fanyi.baidu.com/#zh/en/蜻蜓谷
https://fanyi.baidu.com/#zh/en/蜻谷
Which would be better, more natural sounding in a brand name?
Well...
Your reference (https://qr.ae/pN9BFQ) is good, but I rarely hear people use "蜻" instead of "蜻蜓". People just don't say like that. When you split a Chinese word, it usually turn into many individual words, but sometimes it's meanless. When people look at a word like "蜻", they won't think about the dragonfly, they will feel like it's a bug because the "虫" part. "蜻" itself may have some legendary magic feelings. If you want to convey that kind of feeling to you brand, you can do it.
I would say it is 蜻蜓谷.
The situation for 蝶 and 蜻 are different in modern Chinese. In 《现代汉语词典》, 蝶 does have any entry with the explanation 蝴蝶, but 蜻 does not an entry for 蜻蜓.1 This shows that in modern Chinese, 蝶 can be used for 蝴蝶, but 蜻 cannot be used for 蜻蜓.
Also, there is a website which gives some example words with 蝶 and 蜻. There quite some words in which 蝶 has the meaning 蝴蝶, such as 蝶恋花, 招蜂引蝶. But for 蜻，they are all multi-syllable words where only the word as a whole refers to some kind of insect, such as 蜻蜓，蜻蛉.
蜻 means dragonfly in classic Chinese, but the character cannot be used alone in modern Chinese.
However, I cannot comment whether it can be used in a brand name. (Names are a bit different from normal text)
（1) 蝴蝶 is the only entry for 蝶. 蜻 does not have any explanation at all as a single character.
Many Chinese morphemes (smallest unit of meaning) consists of more than one character. And sometimes with characters looking similar which can't be further subdivided. Many single characters considered morphemes in ancient times may become invalid in the modern time.
Words like these are no longer acceptable in modern Chinese if one uses only a single character from these words.
I believe 蜻蜓 has the same situation. The word should exist as 2 characters together. The reason translation services still works is mostly because they are confident that you meant "Dragonfly Valley", not because the particular Chinese source was grammatically correct or common. These services may be using machine learning or traditional natural language processing, which would likely not throw you an error when seeing something unexpected.
蝴蝶 is a modern word. It was called 蝶 in ancient times. 蝶 is not the abbreviation of 蝴蝶. In fact, 蝶 is the archaic form of 蝴蝶.
蜻蜓 is a disyllabic word since ancient times.
They are completely different. You should not shorten 蜻蜓.
It is like, you can quote some Classical Latin terms in English text, but you cannot shorten the word "monkey" to "mon".
