We use Yuuri for Katsuki because that's a more accurate romanization of the kanji for his name. The spelling and pronunciation in Japanese is ‘Katsuki Yuuri’. The first kanji for his name is 勇 (for courage, bravery, etc.) and the onyomi reads as yuu. Thus, you know, "Yuuri" instead of "Yuri".
Meanwhile, with Plisetsky, the Russians spell his name with only one "u" in the first place. Yuri's name in cyrllic is Юрий, which would translate to either Yuri or Yuriy, thus why we use only one u in his name! In general when the Japanese pronounce Юрий/Yuriy, they go with a long "uu" for the middle, thus why its ユーリ and not ユリ. In the end though, "Yuri" is the most accurate spelling we can use while staying true to what's technically official.
Yuratchka is not Yuri Plisetski's first name, it's just "Yuri" . Yuratchka is simply an affectionate/pet version of his name used by his Grandfather and Yakov (who are both very fond of him). Yuri/Yuratchka in English would be equivalent to George/Georgie.
Pet/affection names are very common in Russia (and often used more than actual names between friends, family and co-workers), Notice that coach Yakov typically calls Victor "Vitya"