Ouran Public High School is mentioned in the anime episode, Beware the Physical Exam! While it is never seen, it is revealed during the episode that the school is where the daughter of an down-and-out physician, Doctor Yabu, is attending.
Dr. Yabu has come to Ouran Academy by mistake and is accused of being a pervert for accosting one of Ouran's students, thinking the girl is his daughter. Dr. Yabu also approaches Haruhi while she's changing in the special clinic set up by Kyoya. When Dr. Yabu is stopped in his tracks by the Ouran Host Club, he reveals a sad story about his wife and daughter leaving him. Kyoya then asks the man if he's looking for Ouran Public High School, since a doctor managing a poor clinic could never afford Ouran's tuition. Tamaki insists that the man be given a map to help him find his daughter.
Ouran Public High School is also mentioned in Episode 2 of the dorama. As in the anime, Doctor Yabu manages a small clinic and is seeking his daughter in order to give her a present for beginning high school. Tamaki helps him and realizes that "a father will always be a father."
The image shown in the article actually depicts Hikarizaka Private High School in the fourth episode of CLANNAD, but is similar to many Japanese high schools called "public" in the US.