is basically correct. In Japanese, "suki" 好き is a noun form of the verb suku 好く, "to like, to love," and is very common in speech. 好き does not exist as a feminine name. It is a misconception fostered by its appearance in numerous "baby name books" and now, websites, over the years, that Suki is a normal Japanese girl's name. For this reason, people of Japanese descent who do not actually speak Japanese might choose it from one of these books as a baby name.
The Japanese Baby Names Project does list three spellings of Suki which have apparently been used as names:
スキ (phonetic katakana spelling)
好希 ("to like" + "to hope")
寿紀 ("longevity" + "history, chronicle")
The fact that only these three can be found indicates that Suki is extremely rare and probably considered something of an aberration in Japan. Regularly used girl's names typically have anywhere from thirty to a hundred kanji spellings.
The word tsuki 月 "moon, month" has no relationship to Suki and does not have the same pronunciation. However, Tsuki does exist as a girl's name, though it is not common. There are 18 kanji spellings included in the Baby Names Project.