China visa for children with one Chinese parent; moving to China with children

This sounds correct. The child born before your wife got permanent residency is a Chinese citizen, while the child born after isn't. According to Article 5 of the PRC nationality law, a child born outside China to at least one Chinese citizen parent has Chinese nationality at birth, unless at least one parent is a Chinese citizen who has settled abroad, and the child has foreign nationality, in which case the child does not have Chinese nationality. The PRC government interprets "settled abroad" has having foreign permanent residency.

Article 5: Any person born abroad whose parents are both Chinese nationals or one of whose parents is a Chinese national shall have Chinese nationality. But a person whose parents are both Chinese nationals and have both settled abroad, or one of whose parents is a Chinese national and has settled abroad, and who has acquired foreign nationality at birth shall not have Chinese nationality.

Both of your children were born to at least one Chinese citizen parent. At the time of the younger child's birth, at least one parent (your wife) was a Chinese citizen who had settled abroad, and the child has foreign nationality (Swedish) at birth; therefore, your younger child does not have Chinese nationality. At the time of the older child's birth, neither parent was a Chinese citizen who had settled abroad (your wife had not "settled abroad", and you were not a Chinese citizen), and therefore your older child automatically has Chinese nationality at birth even though he/she also has Swedish nationality at birth.

See also this page from the PRC embassy in Zambia that contains answers about whether a child has Chinese nationality in various situations. In particular, see (translated from Chinese):

1, I am a Chinese citizen in country X, holding a permanent residence visa in country X, my partner is a country X citizen, our child was born in country X, who has country X nationality at birth according to the nationality law of country X. Now we want to return to China to visit, what document should the child apply for?

According to our country's nationality law, your child does not possess Chinese nationality. They should apply for a Chinese visa.

4, I am a Chinese citizen in country X, holding a short-term visa in country X, my partner is a country X citizen, our child was born in country X, who has country X nationality at birth according to the nationality law of country X. Now we want to return to China to visit, what document should the child apply for?

According to our country's nationality law, your child possesses Chinese nationality. Regarding the situation of conflict of nationality, according to our relevant rules, your child should apply for a Travel Document.

Since your older child is a Chinese citizen, he/she cannot be issued a Chinese visa (unless you first renounce their Chinese nationality). The current PRC policy is to issue such dual-national children a PRC "travel document", a passport-like booklet valid for 2 years which identifies the child as a citizen of the PRC. This travel document can be used in conjunction with their Swedish passport to enter and exit China. If the child takes up residence in Mainland China, they can generally be added to the hukou, with their travel document and a properly authenticated and translated foreign birth certificate (although success may vary depending on locality in Mainland China). If they are on hukou, they should be able to use the benefits of any other Chinese citizen on the mainland.

Since your younger child is not a Chinese citizen, they will need to get a Chinese visit to visit Mainland China. There are visas of various lengths for relatives of people living in China, but I'm not sure if they will be able to access benefits.